Math  104  -­‐  Yu  

Math  104  –  Calculus  10.2  Infinite  Series  

Math  104  -­‐  Yu  

Infinite  series  •  Given  a  sequence  we  try  to  make  sense  of  the  infinite  sum  of  its  terms.  

   

Nicolas Fraiman Math 104

Infinite series• Given a sequence we try to make sense of the infinite

sum of its terms.

• Example: an =1

2n

s1 = a1 =1

2

s2 = a1 + a2 =1

2+

1

4= 0.75

s3 = a1 + a2 + a3 =1

2+

1

4+

1

8= 0.875

s8 = a1 + · · ·+ a8 =1

2+ · · ·+ 1

256= 0.996

s20 = 0.99999905

Math  104  -­‐  Yu  

Infinite  series  

Nicolas Fraiman Math 104

Infinite series

Math  104  -­‐  Yu  

Geometric  series          

Nicolas Fraiman Math 104

Geometric series• A geometric series is one in which each term is obtained

from the preceding one by multiplying it by the common ratio r.

1X

k=1

arn�1 = a+ ar + ar2 + ar3 + · · ·

Nicolas Fraiman Math 104

Geometric series• A geometric series is one in which each term is obtained

from the preceding one by multiplying it by the common ratio r.

• Does not converge for some values of r

1X

k=1

arn�1 = a+ ar + ar2 + ar3 + · · ·

r = 1 then1X

k=1

arn�1 = a+ a+ a+ a+ · · · ! 1

r = �1 then1X

k=1

arn�1 = �a+ a� a+ a� a · · ·

Nicolas Fraiman Math 104

Geometric series• A geometric series is one in which each term is obtained

from the preceding one by multiplying it by the common ratio r.

• We have

1X

k=1

arn�1 = a+ ar + ar2 + ar3 + · · ·

sn = a+ ar + ar2 + · · ·+ arn�1

rsn = ar + ar2 + ar3 + · · ·+ arn

sn � rsn = a� arn

sn =a(1� rn)

1� r

Math  104  -­‐  Yu  

Geometric  series        

Nicolas Fraiman Math 104

Geometric series

Math  104  -­‐  Yu  

Geometric  series          

Nicolas Fraiman Math 104

Geometric series

1X

n=1

1

4n=

1

3

Math  104  -­‐  Yu  

RepeaEng  decimals        

Nicolas Fraiman Math 104

Repeating decimals• We can use geometric series to convert repeating

decimals to fractions.

• Example: 1.23 = 1.2323232323 . . .

1.23 = 1 +23

100+

23

10000+

23

1000000+ · · ·

1.23 = 1 +1X

n=1

23

100n= 1 +

23/100

1� 1/100

= 1 +23/100

99/100= 1 +

23

99=

122

99

Nicolas Fraiman Math 104

Repeating decimals• We can use geometric series to convert repeating

decimals to fractions.

• Example: 1.23 = 1.2323232323 . . .

1.23 = 1 +23

100+

23

10000+

23

1000000+ · · ·

1.23 = 1 +1X

n=1

23

100n= 1 +

23/100

1� 1/100

= 1 +23/100

99/100= 1 +

23

99=

122

99

Math  104  -­‐  Yu  

Telescoping  series          

Nicolas Fraiman Math 104

Telescoping series• A telescoping series is one in which the middle terms

cancel and the sum collapses into just a few terms.

• Find the sum of the following series: 1.2.3.

1X

n=1

✓3

n2� 3

(n+ 1)2

◆

1X

n=1

3

k(k + 3)

1X

n=1

✓1

ln(n+ 2)� 1

ln(n+ 1)

◆

Nicolas Fraiman Math 104

Telescoping series• A telescoping series is one in which the middle terms

cancel and the sum collapses into just a few terms.

• Find the sum of the following series: 1.2.3.

1X

n=1

✓3

n2� 3

(n+ 1)2

◆

1X

n=1

3

k(k + 3)

1X

n=1

✓1

ln(n+ 2)� 1

ln(n+ 1)

◆

Math  104  -­‐  Yu  

Divergence  Test          

Nicolas Fraiman Math 104

Divergence test

Nicolas Fraiman Math 104

Divergence test!

!

Proof: an =nX

k=1

ak �n�1X

k=1

ak = sn � sn�1 ! L� L = 0.

Nicolas Fraiman Math 104

Divergence test!

!

Proof: an =nX

k=1

ak �n�1X

k=1

ak = sn � sn�1 ! L� L = 0.

Math  104  -­‐  Yu  

Examples          

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Examples1. Does converge?2. Does converge?

!

1X

n=1

3n2

n(n+ 1)

1X

n=1

sin(n)

Nicolas Fraiman Math 104

Examples1. Does converge?2. Does converge?

!

Remark:

The converse is not true but diverges!

1X

n=1

3n2

n(n+ 1)

1X

n=1

sin(n)

1X

n=1

1

n1

n! 0

Math  104  -­‐  Yu  

ProperEes  of  convergence          

Nicolas Fraiman Math 104

Properties of convergent series

Nicolas Fraiman Math 104

Properties of convergent series!

!

!

!

!

Example:1X

n=1

3

2n= 3

1X

n=1

1

2n= 3 · 1 = 3.

Math  104  -­‐  Yu  

Reindexing  series