IB Math HL Series Summary

Summary of Series Convergence/Divergence Theorems A partial sum sn  is defined as the sum of terms of a sequence  { ak }. For instance, n

a s  = n

k

=  a 1  +  a2  +  a3  + · · · + an 1  +  a n −

k=1

If the limit of the partial sums converge as n  → ∞  then we say the infinite series converges and write: write: ∞

 s  = a   = lim s k

n→∞

n

k=1

If the limit limn

→∞

  then the series

∞

sn  does not exist then we say the infinite series

(GST) Geometric Series Test If  an = a r n 1 series which converges only if  |r|  <  1 in which case −

∞

a r n=1

a r =

n

 ak  diverges.

∞

n=1

∞

n−1

k=1

=  a  + ar  +  ar 2 + · · ·  =

n=0

a

,

1−r

 an   is a geometric

|r|  <  1

If  | | r| ≥  1 the geometric series diverges. (PST) P-Series Test  If  p >  1 the series  If limn (DT) Divergence Test Test If

→∞



1 np

  converges. If  p  ≤  1 the series diverges.

an   = 0 then the series

∞



n=1

 an   diverges.

(IT) Integral Test  Suppose an = f (n) where f (x) is a positive continuous function which decreases for all x  ≥  1. Then ∞

a

 

∞

n

and

f (x)dx

1

n=1

either either both b oth converge converge or both diverge diverge.. The same holds if  f (x) decreases for all x > b  where b  ≥  1 is any number. (CT) Comparison Test Let Test  Let N  be some integer N  ≥ 1. Then 1) if 0  < an  ≤  b n  for all n  ≥  N  and 2) if 0  < bn  ≤  a n  for all n  ≥  N  and (LTC) Limit Comparison Test lim

n→∞

Then either both series

a

n

and

 

∞

 b conve converge rgess n=1 n

then then

∞

then then

n=1

an bn

 bn div diverge ergess

=  c >  0

 b  converge or they both diverge. n

 

∞

n=1 ∞

n=1

 an  converges  an   diverges

(AST) Alternating Series Test Suppose an > 0 an+1 ≤ an

for all n

lim an = 0

n→∞

then the alternating series ∞

(−1)

n−1

an  =  a 1  − a2  +  a3  − a4  +  a5  − a6  + · · ·

n=1

converges. (ASET) Alternating Series Estimation Theorem Suppose  s  = n=1 (−1)n 1 an  is a convergent alternating series satisfying the hypotheses of (AST) above. Then the difference between the nth partial sum and s  is at most the value of the next term an+1 , i.e.,



∞

  (−1)  ∞

n−1

  a  − s  =  | s − (a  − a  +  a  − a  + · · · + a )| ≤  a  n

n=1

n

1

2

3

4

n

−

n+1

The last two tests for series convergence require the following definitions:

 |a | converges then  a  is said to be absolutely convergent.   |a | diverges then  a  is said to be conditionally convergent. If  a  converges but If 

n

n

n

n

n

Note that (by a Theorem) every absolutely convergent series is convergent.

(RT) Ratio Test 1) If limn

→∞

2) If limn

→∞

3) If limn

→∞

     

an+1 an

an+1 an

an+1 an

     

=  L < 1

then

=  L > 1 =  ∞  

then then

 a  converges absolutely (and hence converges)  a   diverges  a   diverges n

n n

= L = 1 then the test fails and nothing can be said

(RooT) Root Test 1) If limn

→∞

2) If limn

→∞

3) If limn

→∞

 |a |  |a |

=  L < 1

then

=  L > 1 =  ∞  

then then

 |a |

= L = 1 then the test fails and nothing can be said

n

n

n

n

n

n

 a  converges absolutely (and hence converges)  a   diverges  a   diverges n

n n