BSc(H) MATHEMATICS (BOOKLET REAL ANALYSIS)

SECTION-I (REAL NUMBERS & TOPOLOGY OF Question 1

¡

)

State algebraic properties of real numbers.

Question 2 Theorem-: (a) (a) If z and a are elements in ¡ with z+a =a, then z=0. (b) (b) If u and b { 0 are elements in ¡ with u.b=b,then u=1 (c) (c) If a ¡ ,then a.0=0 Question 3 Theorem-: (a) (a) If a { 0 and b in ¡ such that a.b=1 then b=1/a. (b) (b) If a.b=0,then either a=o or b=0 Question 4 Theorem-: There does not exist exist a rational number r such that r 2 =2

Definition-: The order properties of ¡

There is a non empty subset P of ¡ , called the set of positive real numbers That satisfies the following properties: (1) (1) If a, b belong to P,then a+b belongs to P. (2) (2) If a , b belong to P,then ab belong to P. (3) (3) If a belong to ¡ ,then exactly one of the following is true:

a P, a=0, -a P Definition-: let a , b be elements of ¡ . (a) (a) If a-b P ,then we write a>b or b
PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 5 Theorem-: let a , b , c be any elements of ¡ . (a) (a) If a>b and b>c , then a>c. (b) (b) If a>b,then a+c >b+c. (c) (c) If a >b and c>0, then ca>cb.

If a> b and c<0, then ca0.

Theorem-: (a) (a) If a

(b) (b)If n ¥ , then n>0. Question 7 Theorem-: If a ¡ is such that 0 e a< I for every I >0, then a=0. Question 8 Theorem-: If ab>0, then either

(1) a>0 and b>0 or (2) a<0 and b<0 Question 9 (a) (a) Let a u o and b u 0.Then

a < b

2

a b

2

a b

(b) If a and b are positive real numbers, then a.b e

1 2

( a  b)

INEQUALITY). if (c) ,(BERNOULLIS INEQUALITY)

n x " 1, then 1 x  u nx,  n ¥


Question 10

If a , b are real numbers then prove the following. (a) If a+b=o, then a+b=0 ,then b=-a (b) (-a)=a (b) (-a)=a (c) (-1)a=-a, (d) (-1)(-1)=1 (e) (a+b)=(-a)+(-b) (e) (a+b)=(-a)+(-b) (f) (a).(-b)=a.b (g)

1 !  ¨© 1 ¸¹ a ©ª a ¹º

Question 11

If a

¡ satisfies a.a=a, prove that either a=o or a=1.

Question 12

if a if a { 0

¨ 1 ¸ ¨1 b { 0 ,show that. ©© ¹¹ ! ©© ª ab º ª a

¸¨ ¸¨ 1 ¸ ¹© ¹© ¹ ¹© ¹© b ¹ ºª ºª º

Question 13

Show that there does not exist a rational number s such that s 2 =6 Question 14

Show that there is no rational whose square is 3. Question 15

Show that sum and product of two rational numbers, is a rational number. Question 16

(a) Prove that if x is rational and y is irrational then x+y is irrational. (b) If x is non zero rational and y is irrational then x.y is irrational. (c) Show that there is no natural number between 0 and 1 (d) Prove that no natural number can be both even and odd PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

(e) If c>1 and m , n are natural numbers ,show that c

m

"c

n

(f) If 0
n.

" m m

"n

Question 17 (a) (a) If a0 , then

1

a

>0 and

1

¨1¸ © ¹ ªaº

!a.

1

(d) (d) Show that if a
Question 18 (a) (a) If 0
There is a set

¥

 c 1

 c  c2 .

and n ¥ , showthat a  b

¥

an  bn

of natural numbers) numbers)

(called set of natural numbers) which has following properties.

N1. 1 belongs to ¥ N2. if n ¥ , then n 1 ¥ N3.

1 is not succ successor essor o any elemen elementt in ¥

N4 .

i n and and m hav havee the the sam same suc succe cess ssor or then hen n ! m

N5. if S ¥ and has property that (a) 1 belongs to ( b) b) n  n 1 S, then S=¥ PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Axiom N5 is the basis of mathematical induction. MATHEMATICAL MATHEMATICAL INDUCTION-:

P , P , P ............ be statements or propositions, propositions, then 1 2 3 P , P , P ............ are true provided 1 2 3

Let

(I ) P is true, true, 1 1 (I ) P is true whenever P whenever Pn is true. true. 2 n1 Example 1 Prove that 1 2  3  4  ...... n !

n n 1  2

Example 2 All numbers

7n  2n are divisible by 5

Example 3

sin(nx) e n sin( sin( x) for all natural numbers n . Show that sin(

Example 4

12  22  32  42  ...... n2 ! n n 12n 1,  n ¥ 6 Example 5

2 13  23  32  ...... n3 ! 1 2 3..... n  ,  n ¥ PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Example 6 3 11 .....

8n  5! 4n2  n,  n

¥

Example 7 (a) (a) Prove that

1 1 1 1 1   .... n ! 2  n ,  n ¥ 2 4 2 2

(b) (b) Prove that

n n 11    4 is divisible by 7 ,  n ¥

(c) (c) Prove that

7n - 6n -1 i s divi sible by 36,  n ¥

(d) (d) Prove that

2n1 2n3 2n5 ... 4n1! 3n2, n

.

¥

Question State and prove binomial theorem using mathematical induction. Definition-: A number is called an algebraic number if it satisfies a polynomial

equation

an xn  a xn1  ..... a x  a ! 0 , where n1 1 0

a , a , ....,an are integers, integers, an { 0 and n u 1. 0 1

Rational zero theorem. Suppose that

a , a , ....,an are integers and that r is a rational number 0 1

satisfies the polynomial equation

an xn  a xn1  ..... a x  a ! 0 where an { 0 and n u 1. n1 1 0


Write

r ! , p

p, q { 0

are integers having no common factors, factors,

q

Then

q divides an and p divides a . 0

Question

2 can not represent a rational number . Question

17 can not represent a rational number . Question

61/3 can not represent a rational number. Question

1/ 2 3 1 / a ! ¨© 2 5 ¸¹ does not represent a rational number. ª º Question

b!

1/ 2

 2 /7  4

3

can not represent a rational number.

Question (a) (a) Show that (b) (b) Show that

24 and 31 are not rational numbers. 1/4 21/3, 51/7 , and 13 are not rational numbers. 3

,

5,

7,


¨ (c) (c) Show that © 2 ª



1/2

2 ¸¹ º

,

¨ ©5 ª

1/3

¸ 3¹ º

, and

¨ ©3 ª



2/3

2 ¸¹ º

are not

rational numbers.

Absolute value and real line Definition-:The absolute value of a real number a, denoted by

®a ± a ! ¯o ±a °

a ,is defined by

if a "0 ¾ ± if a !0 ¿ if a 0À±

Question 19 Theorem- : (a) (a)

ab ! a b

f or all a, b ¡ .

(b) a 2 ! a2 f or all a ¡ .

(c)if c u 0, then a e c

c e a e c.

(d )  a e a e a f or all r all a ¡ . PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 20 Theorem-:(Triangle inequality) inequality)

If a, b

¡ , then a  b e a  b .

Question 21 Corollary-: If

a, b ¡ , then

(a) a  b e a  b , (b) a  b e a  b . Question 22

let the funct ion f be f be d e fine fined by d by 2  3x 1 2 x (f )x! ofr 2 e x e 3. 2 x 1 find a con stt M such that f ( x) e M  x

« 2, 3» ½

Definition-:

hoo d o f a let a ¡ , and let I " 0.then the I  neig hbour hoo f a i sthe set V (a) ! _x ¡ : x  a  I a I

Question 23 Theorem-:

let a ¡ .if x bel ong s then x ! a .

to the

hoo d V ( a) fo r neig hbour hoo ever y I " 0 I


Question 24

if a , b ¡ and b { 0 then a (a) a ! , b b

sho

that

(b) a ! a2

Question 25

if a, b ¡ , sho

that

ab ! a  b

a.b u 0

Question 26

if x, y, z ¡ and x e z show int er pr et thi s g eomet rically rically.

that

xe ye z

x  y  y  z ! x z

Qestion 27

S how how that

xa I

a  I  x  a  I .

Question 28

if a  x  b and a  y  b , sho

that

x y  b a.

Question 29


Find all x ¡ (a)

y the that sat i s f

4x  5

e 13

f oll o ing ine inequall it ie s. (b) x2 1 e 3.

(c) x 1  x  2 ! 7 (d ) x 1 " x 1 . (e) x  x 1  2

( f ) 4  x  2  x 1  5

Question 30

and sk et ch the points  x, y  in ¡ 2 (a) x ! y (b) x  y !1 (c) x. y ! 2 (d ) x  y ! 2 (e) x e y ( f ) x  y e1 ( g ) x. y e 2 (h) x  y u 2

Determine

that sat i s f y :

Question 31

let I " 0 and H " 0, and a ¡ . S ho V (a) I V (a) and V (a) UV (a) I H I H

that

hoo d s o f a f or some a pp r o pria ar e K  neo g hbour hoo riat e val ue s o f K

Question 32

let a, b ¡ , such that a { b , S ho I " 0 such that V I (a) I V I (b) ! .

that ther e

e xi sts


Question 33

S ho

that if a, b

¡

then

(a) max _a,ba! 1 a  b  a  b  2 (b) min _a, ba! 1 a  b  a  b  2 (c) min _ a, b, c a! min _min _ a, b a , ca . S u pr emum, infimum

and completness property of ¡

De inition-: let S be a nonempty subset o ¡ (a) The set S is said said to be bounded above above i there exists exists a number u ¡ such that se u or all s S. Each such number u is called an upper upper bound o S.

( b) b) The set S is said to to be bounded belo below i there exists exists a number w ¡ such that w e s or all s S. Each such number w is called a lower lower bound o S. (c) A set set is said said to be bounded i it is both bounded above and bounded below. below. A set set is said said to be unbounded unbounded i it is not bounded. bounded.


De finit ion :

let S be a nonempty subset of ¡ . (a) If S is bounded above, above, then a number u is said to be supremum (or a least upper bound) bound) of S if it satisfies the conditions: conditions: S,and (1) u is an upper bound of S, (2) if v is any upper bound of S, S,then u e v. ( b) b) If S is bounded belo , then a number is said to be an infimum (or a gre greatest test lo er boun bound d) of S if it satisfies the conditions: onditions: (1) is a lo er bound of S,and (2) if t is is any any lo er bou bound nd of S, S, then t e .


lemma-:A lemma-:A number u is the supremum of a nonempty subset S of ¡ iff u satisfies the conditions conditions:: (1) s e u  s S, (2) If v0  an sI S such that uu-I

Question 35 let S be any nonempty subset of ¡ that is bounded belo . prove that infS=infS=-sup _ s : s S a. Question 36 if a set S ¡ contains one of its upper bounds, sho that this upper bound is the supremum of S. S. Question 37 let S ¡ be nonempty. sho that u ¡ is an upper bound of S if and only if the conditions t ¡ and t " u  t S. Question 38 let S ¡ be nonempty.sho that u ! supS . (1)  n ¥ , u  1 is not an upper bound of S n (2)  n ¥ , u  1 is an upper bound of S. n Question 39 Sho that if A A and B are bounded subsets of ¡ . then A B is a bounded subset of ¡ . sho that sup(A B) ! sup( sup(supA, sup A,supB supB). ). Question 40 let let S be a bounded bounded subset subset of ¡ and let let S be a nonempt nonempty y subset subset of S 0 then infS e infS e sup S e sup S. 0 0 Question 41 let S ¡ and suppose that s* ! supS,and that s* ¡ . if u S. sho

®

¾

± °

À±

that supS _ua! sup¯±s*, u ¿±.

Question 42 Sho Sho that a nonempty finite set S ¡ contains its supremum. PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 43 Archimedean property-: property-: i x ¡ , there exists n x ¥

such that n x " x. Question 44 

Corollary-: orollary-: i S !



®1 ¯ :n

n

¡



¾

¥ ¿ , then in S ! 0 

À

Question 45 Corollary-: orollary-: i t " 0, there exists n

t

¥ such that

0< 1 < t. nt Question 46 Corollary-: orollary-: i y " 0 ,there exists n

y ¥ such that n y1e y e n . y

Question 47

There exists a positive real number x such such that x 2 ! 2.


Question 48 Theorem-:( Theorem-:(Density Density theorem) theorem) ber s x  y, then ther e if x and y ar e an y an y r eal number e xi sts rat ional nal number ber r ¤ such that x  r  y. Question 49 Corollary-: orollary-: if x and y ar e r eal number ber s with x  y, then ther e e xi sts an irrat ional nal number ber z such that x  z  y. Question 50 ® ¾± ± 1 (a) show that sup ¯1 : n ¥ ¿ ! 1 ± n ° À± (b) if

® ±1

¾

± °

À±

S ! ¯

1 : n, m ¥ ±. find inf S and sup S .  ¿ n m

Question 51 let S be a nonempty bounded set in ¡ . (a) let a " 0, and le let aS ! _a s : s S a. pr ove

that

inf (aS ) ! a inf (S ). sup( sup(aS ) ! a sup( sup(S ). (b) let b  0, and let bS ! _b s: s S a. pr ove that inf (bS ) ! b sup( sup(S ), sup( sup(bS ) ! b inf S . Question 52 let X be nonempty set and let f : X p ¡ have ave bound ed rang e in ¡ . if a ¡ , show that sup a_ f ( x) : x X ! aa  sup _f (x) : x X a al so so

that

inf a_ f ( x) : x

X

! aa  inf _f (x) : x

X

a

Question 53 let A and B be bounded nonempty subsets of ¡ . and let A  B ! _a  b :a A and b Ba. pr ove that sup(A+ sup(A+B B) ! sup( sup( A)  sup( sup(B )

and inf ( A  B) ! inf ( A) inf (B )


Question 54  Given any x ¡ , show that Question 55 if y " 0, show that  n ¥ Question 56 if a " 0, then then sho such that z 2 =a =a..

a unique n ¢ such that n1e x  n. such that

1  y. 2n

that that there there exists exists a posi positive tive real real number number z

Question 57 sho that  a positiv positivee real real number number u such that u3 ! 2. Question 58 if u " 0 is a real number and x  y , sho that  a rational number r such that x ru y. Question 59 zat ion theorem)-: theorem)-: (C haract eri za if S is a subset subset of ¡ that contains contains at least least t o points points and has the property if x , y S and x  y then « x, y½» S, then S is an interval. interval.


Definition-: efinition-:

A neighbourhood of a point x ¡ is any set V that contains an I -neighbourhood VI ( x ) of x for some I " 0.

Definition-: efinition-:

(1) A subset G of ¡ is open in ¡ if for each x there exists a neighbour hood V of x such that V G. (2) A subset F of ¡ is closed in ¡ if the complement C(F) ! ¡ \ F is open in ¡ . Question 60 properties-: O pen set properties-: (a) The union un ion of arbitrary collection of open subsets in ¡ is an open set. set. ( b) b) The intersection of any finite collection collection of open sets in ¡ is an open set. set.

G

Question 61 Closed set properties-: properties-: (a) The intersection intersection o an arbitrary arbitrary colle collection ction o closed sets in ¡ is a closed set. set. ( b) b) The union o inite colle collectio ction n o close closed d sets sets in ¡ is close closed d. Question 62 Theorem-: A subs subseet o ¡ is closed osed i and onl only i it conta ontain inss all it cluster points. points.


Definition-: (interior point)

of A ¡ x ¡ i s calle called d an interior point of A if and only if there exists a neighbourhoo neighbo urhood d V of x such that V A. Definition-:( Definition-:(isolated isolated point) point) x ¡ is called isolated point of A of A ¡ if it is not an interior point of A. of A. Definition-:( Definition-:(closure closure)) let A ¡ be any set, set,then closure of A of A is the intersection of all closed sets containi co ntaining ng A. it is denoted by A Definition-:( efinition-:(interior interior of a set) set) Let A ¡ be any set, set,then interior of is the union of all open sets contained in A. it is denoted by A0. Definition-:( Definition-:(aa) A set S is said to be denumerable denumerable((or countably infinite) infinite) if there exists a bijection of ¥ onto S. ( b) b) A set S is said to be countable if it is either finite or denumerable denumerable.. (c) A set S is said to be uncountable if it is not countable. countable . Question Sho that set ¡ (of real numbers) numbers) is uncountable. uncountable.


SECTION -2(SEQUENCES) Definition-:

A sequence of real numbers (or a sequence in

¡ ) is a function

defined on the set ¥ of natural numbers whose range is contained in

¡ the set of real numbers. Notation-: X ,

xn , xn : n

¥



Definition-: (Fibonacci sequence)

The sequence  f n  defined as ! f  fn f ! 1, f ! 1, and f 1 2 n1 n1 is called the fibonacci sequence. uence. Definition-: A sequence X ! xn  in

nu 2.

¡ is said to converse to x ¡

Or x is said to be a limit of  xn , if for every I " 0 there exists K (I )

¥ such that xn  x  I . for all n u K (I ) .

Note-: If a sequence has a limit we say that sequence is convergent; if it has no limit we say it is divergent. Question 64 PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Theorem-: A sequence in

¡ can have at most one limit.

Question 65 Theorem-: Let X ! xn

  be a sequence of real numbers , and let x

¡ .

The following statements are equivalent (a) X converges to x (b) for every I " 0 ,there exists K (I ) such that xn  x

 I ;  n u K (I )

(c)for every I " 0 ,there exists K (I ) Such that

¥

x I  xn  x I

¥

 n u K (I )

(d)for every I -neighbourhood V ( x) of x ,there exists K (I ) I Such that

xn

¥

VI ( x)  n u K (I ) .

Question 66 Show the following ¨1¸ (a) lim © ¹ ! 0 © n¹ ª º PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

¨ 3n  2 ¸ ¹ © n 1 ¹ ª º

(b) lim © (c) if

!3

0  b 1, then lim( lim(bn ) ! 0

Definition-: if X !

se quence x1, x2 , x , x ,...........xn ............. is a sequence 3

4

Of real numbers and if m is a given natural number , then the m-tail of X is the sequence s equence X

m

! xm1, xm 2 , xm3 , xm 4 ,.........xm n ...........

Question 67 Theorem-: m-tail of a sequence s equence is convergent if and only if the sequence is convergent, moreover both shall have the same limit. Question 68 Theorem-: Let

 xn be a sequence of real numbers and let x

If an  is a sequence s equence of positive real numbers with and if for some constant C " 0 and some m

xn  x e Can then it follows that

¡ .

lim an ! 0

¥ we have

 num ,

lim xn ! x . PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 69

¨ 1 ¸ if a " 0, then lim © (a) if a !0 ¹ ª 1 na º ¨ 1¸ if a " 0, then lim © a n ¹ ! 1 (b) if a © ¹ ª º ¨ 1¸ (c) lim © n n ¹ ! 1 © ¹ ª º Question 70 For any

¨

¸

b ¡ , pr ove that lim ©© b ¹¹ ! 0 ªnº

Question 71 Use the definition of limit of a sequence to establish the following limits. ¨ n (a) lim © © 2 ªn

¸ ¹!0 1 ¹º



¨ 3n (c) lim © © 2n ª

1 ¸¹ ! 3  5 ¹º 2

¨

2n ¸¹ ! 2 © n 1 ¹ ª º

(b) lim ©

¨ 2 1 ¸¹ 1 n © (d) lim © ! ¹ 2 2 © 2n 3 ¹º ª




Question 72 Show that ¨ (a) lim © © ª ¨ © (c) lim © © © ª

1 ¸¹ ! 0 n  7 ¹º n

¨

n ¸¹ ! 0 © n 1 ¹ ª º

(b) lim ©

¸

1 n ¹ n2 1

¹!0 ¹ ¹ º

Question 73

1 f or n ¥ . ln( ln(n 1) (a) use the defini definiti tion on of limi limitt to sho sho that that lim lim ( xn ) ! 0 (b) find a specific value of k (I ) as required in the definition of limit for each of (1) I ! 1 (2) I ! 1 2 10

let xn !

Question 74

f or a sequence uence o real real number numberss (xn ) sho

that lim( lim(xn ) ! 0

lim xn ! 0

give an example to show that convergence o  xn  does not imply convergence o ( xn ). Question 75 PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Show that if xn u 0

n ¥

,and

lim( lim( xn ) ! 0 , then lim

 x ! 0 n

Question 76

Prove that if lim( lim( xn ) ! x , and if x " 0 , then there exists a natural number M such that xn " 0  n u M. Question 77 ¨

1

¸

Show that lim © n ¹ ! 0 ©3 ¹ ª

º

Question 78

Show that for 0  b  1 , lim nbn



! 0 .

Question 80 If lim

 xn ! x " 0, show that  k

such that

¥

i n u k , then 1 x xn  2 x 2

Definition-: A sequence X ! xn  of real numbers is said to be

bounded if  M " 0 such that

xn e M f or all r all n ¥ .

Question 81 Theorem-: Every convergent sequence of real numbers is bounded.


Question 82 Theorem-: (a) Let X ! xn  and Y ! yn  be sequences of real numbers that conversge to x and y , respectively, and let c

¡ .Then the

sequences X  Y , X  Y , X .Y and c X converge to

x  y, x  y, x y and cx , respectively. (b) If X ! xn  converges to x and Z ! zn is a sequence of nonzero real numbers that converges converges to z { 0 , then the sequence

X Z

converges to

x z

.

Question 83 Theorem-: If X ! xn  is convergent sequence of real numbers and if x

n u0

for all n

¥ , then x ! lim xn u 0

Question 84 Theorem-: If X !

xn and Y ! yn  are convergent sequences of real numbers and if

xn e yn

for all

n ¥ , then lim  xn e lim yn .


Question 85 Theorem-: If X !

xn is convergent sequence and if

then

a e lim( lim( xn ) e b .

a e xn e b for all n ¥ ,

Question 86 Theorem-:(squeeze theorem) Suppose that X ! xn , Y ! yn ,

and Z= Z= zn  are such that

xn e yn e zn f or all r all n ¥ , and that lim xn ! limzn  then Y ! yn  is convergent and lim  xn ! lim  yn ! lim zn . Question 87 ¨

Show that

¸

sin(n) ¹ ! 0 lim © sin( © n ¹º ª

Question 88 Theorem-: let the sequence X !

xn  converges to x . Then the sequence  xn 

converges to x .

Theorem-: PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Let X ! xn  be a sequence of real numbers that converges to x and suppose xn u 0 f or all n ¥ , then the sequence

nverg e s and lim  xn conve

xn ! x .

Theorem-: Let X ! xn  be a sequence of positive real numbers such that

l!

¨ x ¸  n 1 ¹ lim ©© © xn ¹¹ ª º

e xi sts. if l  1,

then

lim xn ! 0

Question 89 Find

limit of the following sequences:

(a)

¨ 2 ¸¹ ¸ ©¨ lim © ©© 2  1 ¹¹ ¹ nº ¹ ©ª ª º

¨ n 1 ¸ (c) lim © © n 1 ¹¹ ª º

¨ (1)n ¸ (b) lim © © n  2 ¹¹ ª º (d)

¨ n 1 ¸ lim © ¹ n n ª º

Question 90

if (bn ) i s a bound ed sequence nce and lim an ! 0 then lim( lim(an.bn ) ! 0

Question 91


let yn ! n 1  n f or all n ¥ . sho that ( yn ) and (n yn ) conve nverg e. find their lim its. Question 92 Determine the following limits. ¨ (a) lim © 3 © ª

1/2n ¸¹



n

¹ º

¨ (b) lim © © ª

1/ln( ln(n1) ¸¹

n 1

¹ º

Question 93

if 0  a  b,

  bn1 ¸¹ . ¹ n  b ¹º

¨ n 1 determ determine ine lim lim©© a n a © ª

Question 94

if 0  a, 0  b show that

¨ a b ¸ ¨ ¸ lim © n  a n  b  n ¹ ! © ¹ ª º 2 ª º Question 95 Use the squeeze theorem to determine the limits of following. ¨ 2¸ (a) © n1/ n ¹ © ¹ ª º

¨ (b) ©© © ª

1/n2 ¸¹

n!

¹ ¹ º

Question 96


1/n zn ! an  bn  where 0  a  b, then lim zn ! b sh o

that i

Question 97

if 0  a  1, and b " 1, then ind the limi limitt o ollo ollowin wing g sequences, uences, ¨ bn ¸ n (1) (a ) (2) ©© n ¹¹ 2 ª

¨ ¸ (3) © nn ¹ ªb º

(4)

¨ 3n ©2 ©© 2n ª3

º

¸ ¹ ¹¹ º

Question 98

(a)give (a)give an example of a convergent sequence

( xn ) o f posit ive ive r eal eal nu mber ber s

¨ x ¸  n 1 © ¹ ith lim © © xn ¹¹ ª º

!1

(b)give example of a divergent sequence with same property. Question 99

let ( xn ) be a sequence of positive real numbers such that ¨ ¸ lim © xn1/n ¹ ! l  1. show that  r with 0  r 1 such that 0  xn  r n ª

º

for all sufficiently large n

¥ . use this to show that lim  xn ! 0 .

Question 100


Suppose that  xn  is a convergent sequence and  yn  is such that for any I " 0 there exists M such that xn  yn  I  n u M. Does it

nverg ent ??. follow that ( yn ) i s conve Definition-:

let X ! ( xn ) be a sequence of real numbers. We say that X is increasing if it satisfies the inequalities

x1 e x2 e x3........ xn e xn 1 e ............. We say it is decreasing if it satisfies the inequalities

x1 u x2 u x3........ xn u xn 1 u ............. We say it is monotone if it is either increasing or decreasing. Question 101 Monotone convergence theorem-:

A monotone sequence of real numbers is convergent if and only if it is bounded. Further : (a)if (a)if X ! ( xn ) is a bounded increasing sequence, then

lim( lim( xn ) ! sup _xn :n ¥ a (b)if Y ! ( yn ) is a bounded decreasing sequence, then

lim( lim( yn ) ! in _yn :n ¥ a Question 102 PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

¨

Show that lim © © ª

1 ¸¹ ! 0 n ¹º

Question 103

Let xn ! 1

1  1  1  ...... 1  .. ....... show that ( xn ) is n 2 3 4

unbounded and hence is divergent. Question 104

Let Y ! yn  be defined inductively by

y ! 1, y ! 1 2 yn  3 or n u 1. Show that lim  yn ! 3 . 1 n1 4 2 Question 105

let Z ! zn  be the sequence of real numbers defined by

z ! 1, z ! 2 zn or n u 1.show that lim  zn ! 2 . 1 n1 Question 106

Let a " 0 , construct a monotone sequence of real numbers that converges to

a.

Question 107

Let

n ¸ 1  n ¹¹ for n ¥ . Show that E= (en ) is bounded and º

¨ en ! ©©1 ª

increasing sequence. Also show that 2 e lim( lim(en )  3 . Question 108


let x ! 8 and x ! 1 xn  2 f or n ¥ . 1 n1 2 sho that xn  i s bound ed and monotone. find the limit. limit. Question 109

! 2  1 for n ¥ . Show that  xn  is bounded let x "1 and x xn 1 n1 and monotone. Find its limit. Question 110

let

1

u x 2 and

x ! 1 xn 1 or n¥ . Show that  xn  is n1

decreasing and bounded below by 2. Find limit. Question 111

let

1

x! 1 and

x ! xn  2 n1

or n¥ . Show that  xn  is

convergent find its limit. Question 112

let y ! 1

p

,

p"0

and y ! yn  p f or n ¥ . show that  yn  n1

is convergent. Find its limit. Question 113

let y " 0 , 1

p"0

and y ! yn  p f or n ¥ . show that  yn  is n1

convergent . find its limit. Question 114 PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

let y ! a " 0 , and y ! yn  1 f or n ¥ . determine if  yn  is 1 n1 yn convergent or divergent. Question 115

let A be an infinite subset subset of ¡ that is bounded above and let

u ! sup( sup( ) Show that there exists an increasing sequence  xn 

 ! u .

such that lim xn Question 116

 

Establish the convergence or the divergence of the sequence xn Where xn !

1  1  1  .........  1 f or n ¥ . n 1 n  2 n  3 2n

Question 117 Let xn !1

1  1  ......... 1 f or n ¥ . prove that ( x ) is n 22 32 n2

increasing and bounded, and hence convergent.


Definition-: let X ! xn  be a sequence of real numbers and

n  n  n  .... ...... ..n .... ...... .... .... .... be a strictly increasing sequence of 1 2 3 k natural numbers. Then the sequence ¨ © © ª

xn , xn ,... ,..... .... .... .. 1 2

¸ xn , .... ...... .... .... .... .... .... ....¹¹ k º

¨ ¸ © X d ! © xn ¹¹ k º ª

given by

is called a subsequence of X .

Question 118 Theorem-: if a sequence X ! xn  of real numbers converges to a real ¨

¸

number x , then any subsequence X d ! ©© xn ¹¹ of X ª

k

º

also converges to x . Question 119 Theorem-: let X ! xn  be a sequence of real numbers .

Then the following are equivalent. (a)The (a)The sequence X ! xn  does not converge to x (b)there exists an I 0 " 0 such that

 k ¥ ,  n

k

¡ .

¥ such that

n u k and xn  x u I 0 k k

  f

! xn (c)there exists an I 0 " 0 and a subsequence X d

o

X

k

Such that xn

k

 x u I 0  k ¥

. PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Divergence criteria-: if a sequence X ! xn  of real numbers has either

of the following properties, then X is divergent.

! ( xn ) and (1) X has two convergent subsequence X d k

d X d ! ( xr ) whose limits are not equal. k (2) X is unbounded. Question 120 Monotone subsequence theorem-:

If X ! xn  is a subsequence of real numbers, then there is a subsequence of X that is monotone. Question 121 The Bolzano-Weierstrass theorem

A bounded sequence of real numbers has a convergent subsequence. Question 122 Theorem-:

Let X ! xn  be a bounded sequence of real numbers and let x

¡

have the property that every convergent subsequence of X converges to x .then the sequence X converges to x .

Question 123

Give an example of an unbounded sequence that has convergent subsequence. PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 124

Establish the convergence and find the limits of the following sequences:

(a)

¨ ©¨ © ©1 ©© ©© ª ª

¨ ©¨ (c) ©© ©1 © ©© ª ª

(e)

2 ¸¹ n ¸

 1 ¹¹ n2 º

(b)

(d)

¨ ©¨ © ©©1 ©ª ª

(f)

¨ 3n ¸ ©¨ 1 ¸¹ ¹¹ © ©1 ¹ 2 n © ©ª ¹ º © ¹ ª º

2 ¸¹ 2 n ¸

 1 ¹¹ n2 º

¨ ¸ 1 /2 n © 3n ¹ © ¹ ª º

 

¹ ¹ ¹¹ º

n¸ ¨ ¸ ¹ 1 ©¨ ¹ ¹ © ©©1 2n ¹º ¹ ©ª ª º

¹ ¹ ¹¹ º

¸ 2  n ¹¹ º

n¸ ¹ ¹ ¹ º

Question 125

Prove that limit of a sequence which has the property that its every subsequence has a convergent subsequence that converges to zero, is zero.

Question 126


let ( xn ) be a bounded sequence and ®

¾

sn ! sup ¯ x : k u n ¿ and S ! in ° k À

n ¥

let

_sn ashow that there exists a

subsequence of ( xn ) that converges to S . Question 127

Suppose that xn u 0  n

n ¸ ¨ ¥ and that lim ©© 1 xn ¹¹ e xi sts. ª

º

nverg e s. Show that  xn  conve Question 128

Show that if  xn  is unbounded, then there exists a subsequence ¨ ¸ x ©© n ¹¹ k º ª

such that

¨ ¸ © 1 ¹ ¹ ! 0. lim © x © n ¹ © ¹ k º ª

Question 129

Let xn be a bounded sequence and let s ! sup xn :n ¥ .

 

Show that if s

_

a

_xn :n ¥ a, then there is a subsequence of  xn 

That converges to s .

Definition-: (Cauchy sequence) PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

A sequence X ! xn

  of real numbers is called Cauchy

If 

I "0 

H (I )

¥ such that  n,

m

¥ and n, m u H (I )

We have xn  xm  I . Lemma -: if X !

xn  is a convergent sequence of real numbers then it

is a Cauchy sequence. Lemma -: Every Cauchy sequence is bounded. Cauchy convergence criterion-: A sequence of real numbers is

convergent if and only if it is Cauchy sequence. Question 130

Let X ! xn  be defined by ¸ x x ! 1 x ! 2 and xn ! 1 ¨© x ¹ for n " 2 , discuss the   1 2 2 ª n 2 n 1º

convergence. Question 131

Let X ! xn be defined as

1 1 1 x ! x !  1 1! 2 1! 2!

and

1 1 1 (1) n1 xn !    ............. . 1! 2! 3! n!

Discuss the convergence

Question 132

Give an example of a bounded sequence that is not a Cauchy sequence. PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 133

Show directly from the definition that the following are Cauchy sequences. (a)



¨ n 1¸ © ¹ © n ¹ ª º

(b)

¨ ©1 © ª

¸ 1 1 1 1     ............ ¹¹ 2! 3! 4! n!º

Question 134

Show by definition that sum and product of two Cauchy sequences is again Cauchy sequence.

Question 135

If 0  r  1 and x

n1

 xn  r n  n ¥ , show that  xn  is a

Cauchy sequence Question 136

if x  x ar are ar bit rar rar y r eal number ber s and xn ! 1 ¨© x  x ¸¹ 1 2 2 ª n2 n1 º for n " 2 ,show that  xn  is convergent. What is its limit??. Question 137

1 2  x if x  x ar e ar bit r r ar y r eal eal number ber s and xn ! x 1 2 3 n 1 3 n  2 show that  xn  is convergent. convergent. What is its limit??. Definition-:limit (superior and limit inferior)

(a) Let  xn  be a sequence of real numbers and PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Let U

N

! sup _xn : n u N athen limit superior of the sequence is

limsup( xn ) ! lim( lim(U ) . defined as limsup( N (b)let ( xn ) be a sequence of real numbers and Let V

! inf _xn : n u athen limit inferior of the sequence is

lim(V ) . Defined as limin ( xn ) ! lim( N Definition. For

a sequence  sn , we say that lim( lim( sn ) ! g if and only if

 M " 0  N ¥

such that n " N  sn " M .

Also we say that lim( lim( sn ) ! g if and only if

 M  0  N ¥

such that n " N  sn  M .

Question

¨ n 2 3 ¸ ¹ ! g Show that lim © © n1 ¹ ª º

Theorem-: Let  sn  and t n be sequences such that

lim sn ! g PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

and

lim t n " 0 (finite or g ) , Then lim snt n ! g .

Theorem-: For a sequence  sn  of positive real numbers, we have

¨ 1 ¸ lim sn ! g if and only if lim © ¹ ! 0 . ª sn º Theorem-: (1) If  sn  is an unbounded nondecreasing nondecreasing sequence, then

lim sn ! g

(2) If  sn  is an unbounded nonincreasing nonincreasing sequence, then

lim sn ! g

Corollary-: If  sn  is a monotone sequence, then the sequence either converges, diverges to

g ,or diverges to g .

Definition


Let  sn  be a sequence of real numbers. A subsequential s ubsequential limit limit is any real number or symbol

g or g

that is the limit of some

subsequence of  sn . Theorem-: Let  sn be any sequence s equence of real numbers, and let S denote the set of subsequential limits of  sn . (1) S is nonempty (2)

sup S ! lim sup sn and in

(3)

lim sn exists if and only if

S ! lim in s .

n

e lement , namely S has exacty one element

lim sn . Theorem-: Let S be the set of all subsequential limits of sequence  sn . suppose t n  is a sequence in S

¡ and that t ! lim t n , then t

belongs to S .

Theorem-:


If  sn  converges to a positive real number s and t n  is any sequence then

lim sup sn tn ! slim sup tn .

here we allow the conventions s(g) ! g

and s(g) ! g

Theorem-: Let  sn  be any sequence of nonzero real numbers. Then we have s

lim in

n1 e lim in sn

sn

1/ n

e lim sup sn

1/n

s

e lim sup n1 . sn

Corollary. s

1/ n  n 1 If lim exists, then lim sn exists and both limits are same. sn Theorem-: Let  sn  be any sequence of real numbers (1) If lim sn is defined(as real numbers

g or  g ),

then

lim inf ns ! lim ns ! lim sup ns . (2) If lim inf sn ! lim sup sn then lim sn is defined and lim inf ns ! lim ns ! lim sup ns .

SECTION-3(INFINIE SERIES) PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Infinite series-: An infinite series is an infinite summation of real numbers. It is denoted as

g § an . n!m

Sequence of partial sum-:

g Let § an be an infinite series then sequence of partial sums of this n!m infinite series is the sequence S n  defined as

a  ...........  an ! S n ! am  a m1 m 2

n

§ ak . k !m

Definition-: An infinite series is convergent, divergent according to as

its sequence of partial sums Is convergent or divergent. Question 138

g (a) Discuss the convergence of geometric series § a.r n .

n!0

g 1 (b) Discuss the convergence of p-series § p .

n!1n

Definition-:

we say that a series § an satisfies Cauchy criterion if its sequence of partial sums is a Cauchy sequence. PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Let

§ an be an infinite series, and let (S n )

be sequence of partial

sums of this series. Then in view of above definition we have following.

 I " 0  N ¥ such that n u m " N im pl ie s

n

§ ak  I k !m

Theorem-: A series converges if and only if it satisfies Cauchy criterion. Corollary-: if a series

§ an

is convergent, then lim( lim(an ) ! 0 .

Comparison test-:

Let § an be a series where an u 0 (1)If (1)If § an converges and bn

n ¥

e an  n ¥

, then § bn

converses. (2) If

§ an ! g and bn u an  n ¥

, then § bn ! g

Definition-( Absolutely convergent series)

An infinite series § an is said to be absolutely convergent if and only if the series § an is convergent. Theorem-: Every absolutely convergent series is convergent. Ratio Test.

A series

§ an

of nonzero terms


¨ an1 ¸ (1) Converges absolutely if limsup © 1, ¹ © an ¹ ª º ¨ an1 ¸ "1 (2) Diverges if limin © ¹ © an ¹ ª º ¨ an1 ¸ ¨ an1 ¸ e1e limsup© (3) Otherwise liminf © and the test © an ¹¹ © an ¹¹ ª º ª º gives no information. Root test.

Let

§ an be a series and let E ! limsup a n

The series

1/ n

.

§ an

(1) Converges absolutely if E

1

(2) Diverges if E "1 (3) Otherwise E

!1 and let the test gives no information.

Question 139 Consider the series

g ¨ 1 ¸ n § © ¹ , discuss the convergence.

n !2 ª

3

º

Question 140


Consider the series

¨ ¸ n ¹, § ©© ¹ ª n2  3 º

discuss the convergence.


n § n , check for the convergence. 3


§

¨ ¸ 2 © ¹ n © (1)  3 ¹ ª º

n , check for the convergence.

Question 143

g (1)n n Consider the series § 2 , discuss the convergence. n!0 Question 144 Determine which of the following series converge. Justify your answer

¨ n4 ¸ (a) § © n ¹ ©2 ¹ ª º

¨ 2n ¸ (b) § © ¹ n ! ª º

¨ n2 ¸ (d) § © n ¹ ª3 º

¨ cos2 (n) ¸ g ¨ 1 ¸ ¹ (e) § © ¹ (f) § ©© ¹ © n2 ¹ ª log n º ! n 2 ª º

¨ n 1 ¸ (g) § © ¹ 2 ªn º (j)

§

¨ 3¸ ©n ¹ © n¹ ©3 ¹ ª º

(h) §

(k)

(c)

(1) n

§

¨ 2 ©n © n! © ª

(i) ¸ ¹ ¹ ¹ º

¨ n! ¸ §© 4 ¹ ª n 3 º

§

¨ ¸ 3 n © ¹ © 3¹ ªn º

(l)

§

¨ 1 © © nn ª

¸ ¹ ¹ º


¨ © © ª

¸ 1 ¹ (m) § n! ¹º n ¨1¸ ¨ 2¸ (p) § ©© ¹¹ ©© 50  ¹¹ nº ª 2º ª

¨ 2  cos n ¸ ¹ n ¹ 3 ª º

¨ ¸ 1 © ¹ © ¹ n 2 © ¹   ( n ( 1 ) ) 2ª º

g (s) §

n!

¨

1

¸

(n) § © ©

(0) § © n ¹ © 2 n ¹ ª º

¨ (100) n ¸ (q) § © ¹ n ! ª º

(r)

_

(t) §

n1 n

a

¨ nT ¸ ¹ 9 º

§ sin ©ª

¨ n! ¸ ¹ n ª º

(u) § © n

The integral test-:

§ an

Let

be a series of nonnegative nonnegative terms.

If there is a continuous non-incre non-increasing asing function f : «1, g a ¡ Such that f n ! an  n ¥

¨n ¸ Then series is convergent if lim © ´ f ( x) dx ¹  g ª1 º ¨n ¸ And it diverges if lim © ´ f ( x) dx ¹ ! g ª1 º Alternating series theorem-: If a

1

u a u a u ...... u an u ........u 0 and lim( lim(an ) ! 0 , then the 2 3 n

alternating alternating series § (1) an converges. Question 145

Determine which of the following series converge. justify your answer. PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

(a)

¨ ( 1)n § ©© n ª

g (d) §

n!

¸ ¹ ¹ º

¨ 1 © p ©© n ( log n ) 2ª

(b) ¸ ¹ ¹¹ º

(e)

¨ (1)n n! ¸ ¹ § ©© n ¹ 2 ª º ¨ © © 2 ©ª

g §

n!

g ¨ ¸ 1 © ¹ (g) § © n(log n)(loglog )(loglog n) ¹º n!4 ª

(c) ¸

1 ¹ ¹ n (log n) ¹º

§

« ¨ nT ¸ » ¬ sin © ¹¼ ª 6 º½

n

g (f) §

¨ log n ¸ © ¹ © n ¹ º 2ª

n!

g (h) §

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BSc(H) MATHEMATICS (BOOKLET REAL ANALYSIS)

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