Math 55 3rd Exam Exercises

Math 55 3rd LE  Exercises

+

1. Write rite out the first few terms terms of each each series series to show how the series starts. Then find the sum of the series. +∞ 1  ( 1)n (a) + 2n 5n

(m)

n=1 +

(n)

 −     −   

(b)

∞

n=0

(c)

(o)

 tan−1 (n + 1)

(d)

∞

n=1

(2n (2n

−

n

∞

(q)

6  1)(2n  1)(2n + 1)

∞

n!e−n

n=1 +

(r)

∞

n=1 +

(a)

∞

+

(b)

∞

n=1

2 n n

∞

n=1 +

(d)

∞

n=1 +

(e)

∞

n=1 +

(f) (f )

∞

n=1 +

(g)

∞

n=1 +

(h)

∞

n=1 +

(i)

∞

n=1 +

(j)

∞

n=1 +

(k)

∞

n=1 +

(l)

∞

n=1

n10 10n

2 + ( 1)n 1.25n

+

(t)

∞

1

n=1 +

(u)

1

2n

∞

n=1

ln n n

+

(c)

(s)

  −√       −        √     − 

n=1

 + n  + n

+

2. Determine Determine the conve convergen rgence/ ce/div diverg ergenc encee of  the following series. +

n n

n=1

n=1 +

1

1 1+2+3+ n=1 √  +∞ n 2 (p) 2n

2n+1 5n

tan−1 (n)

10 10n n + 1 n(n + 1)(n 1)(n + 2)

+

+

∞

∞

n=1

n=0 +

∞

  √   ···     −  −      −    

 1

∞

n=1 +

2n

(v)

n + 1

∞

n=1 +

 1 n tan n

(w)

∞

n=1 +

en

(x)

1 + e +  e2n

∞

n=1 +

1

(y)

n3 + 2

∞

n=1

n

3 n

ln n n3 1 n

1 n2

(n + 1)(n 1)(n + 2) n! (n + 3)! 3! n! 3n

n! (2n (2n + 1)!

3. Which of the alternating series below converge and which diverge?

1 ln (ln (ln n)

+

2

(ln n) n3

(a)

∞

( 1)n+1

n=1 +

1 (1 + (ln n)2 )

(b)

∞

( 1)n+1

n=1 +

1  n n 2n

(c)

∞

( 1)n+1

n=1 +

 1 sin n

(d)

∞

( 1)n+1

n=1

1

 n 10

n

−   − −  − √  1 ln n

ln n ln n2

n + 1 n + 1

+

4. Which of the following series converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers. +∞ 1 (a) ( 1)n n

 − √  − − − − − − − −  −  √  − −  − √   −  

(b)

n=0 +

(c)

(b)

∞

( 1)n+1

n=1 +

(c)

∞

( 1)n

n=1 +

(d)

∞

+

(d)

+

(e)

∞

+

(e)

+

(f) (g)

(f)

n

( 1) n (2/3)

(h)

(g)

∞

n=1 +

(h)

∞

(i)

n +1

(j)

n=1 +

(j)

∞

n=1 +

(k)

∞

n=1 +

(l)

∞

n=1 +

(m)

∞

n=1

+

(k)

(n)

∞

( 1)n

n=1

xn

( 1)n+1 (x + 2) n n 2n (x + π)n n (4x

 5)2n+1

n3/2

(b) f (x) =

√ x + 4, a  = 0

7. Find the Maclaurin series expansion of the following functions.

− √ n



√  √  n + n − n

(a) e−5x 1 (b) 1 + x (c) 7cos( x) ex + e−x (d) cosh x = 2 x (e) 1  x (f) cos x

−



+

(a)

∞

n

(a) f (x) = sin x, a  = π/4

5. (a) Find the series’ radius and interval of  convergence. For what values of  x does the series converge (b) absolutely, (c) conditionally?

∞

1 1+ n

6. Find the Taylor polynomials of orders 1, 2 and 3 generated by f  at x = a

( 1)n (2n)! 2n n! n n + 1

n(x + 3) n 5n

n=1

( 1)n (n + 1) n (2n)n

+

(o)

(l)

cosnπ n n

n=1

∞

+

( 1)n−1 n2 + 2n + 1

( 1)n

∞

n=1

( 100)n n!

+

∞

n=1

n=1

(i)

xn n2 + 3

∞

+

2

n ( 1) n + 1

∞

x2n+1 n!

n=1

−1 n

n

+

( 1)n xn n!

+

n tan

( 1)

∞

n=1 n

n=1 +

xn n n 3n

∞

+

n2

2

nxn n + 2

n=0

+

∞

(x  2)n 10n

+

n+1 1 + n

n=1

∞

n=0

5 + n

( 1)

∞

n=0

n+1 3 + n

n=1

∞

n=1

n n3 + 1

1 n + 3

( 1)

∞

n=0

n=1 +

∞

 −   √  −   √     −  √   −

−

( 1)n (4x + 1)n

−√ 

(g) xex x2 (h)  1 + cos x 2 (i) x cos πx x2 (j) 1  2x

−

−

n=0

2

8. Find the Taylor series expansion of the following functions at x  = a. (a) f (x) = x 3 4

ii. Use your answer in the previous item to find the sum S  of the series. +∞ (x  1)n (c) For the power series , 3n n

− 2x + 4, a  = 2 + x + 1, a =  −2



2

(b) f (x) = x 1 (c) f (x) = 2 , a  = 1 x (d) f (x) = 2x , a  = 1

n=1

i. determine its radius of convergence, ii. find the interval of convergence, iii. state clearly whether the series converges absolutely, converges conditionally or diverges at the endpoints of the interval obtained in the previous item.

9.   SAMPLE EXAM (a) Determine whether the sequence/series is convergent/divergent. 1 +∞ − 1 i. tan n n n=1 +∞ 1 ii. tan−1 n n n=1 +∞ 1 n/2 1 iii. + 3 n n=1 +∞ n5 + n4 iv. n7 + n3  n + 1 n=1 +∞ v. e−n

 −  −     √    −      − 

(d) Differentiate the identity 1 3(1

3

vi.

=

∞

3n−1 xn ,

n=0

 | |

−

n=0

∞

− 3x)

+



where x  < 13 , to obtain a power series 1 representation for . (1  3x)2 (e) Find the Maclaurin series expansion of  f (x) = e 2x and then use this to find the +∞ (2ln2)n sum of  . n!

2

+

−



1 n(ln n)2

n=0

series

(f) Consider the function f (x) = ln(1 + x) with x  < 1.

i. Find the nth partial sum s n  of the series.

i. Find the third degree Taylor polynomial of  f  at 0. ii. Use your answer in the previous item to approximate ln(1.5).

n=2

(b) Consider the telescoping ∞ 5 5 . 2k  1 2k + 1 k=1

 | |

−

3