11.2 – Series

http://www.whyu.org/whyUPlayer.php?currentchapter=1&currentbook=3&youtubeid=jktaz0ZautY

http://screencast.com/t/dzyDH0gf

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3

If we try to add the terms of an infinite sequence we get an expression of the form

a1 + a2 + a3 + ··· + an + ∙·∙

which is called an infinite series or just a series. It is denoted, for short, by the symbol

1{ }n na

1

orn nn

a a

4

We consider the partial sums

s1 = a1 s2 = a1 + a2 s3 = a1 + a2 + a3

s4 = a1 + a2 + a3 + a4

◦ In general,

1 2 3

1

n

n n i

i

s a a a a a

5

◦ These partial sums form a new sequence {sn}, which may or may not have a limit.

◦ If exists (as a finite number), then, we

call it the sum of the infinite series Σan

lim

nn

s s

6

Given a series let sn denote its nth partial sum:

If the sequence {sn} is convergent and exists as a real number,

then the series Σan is called convergent and we write:

◦ The number s is called the sum of the series.

◦ Otherwise, the series is called divergent (a divergent series has no sum).

1 2 3

1

nn

a a a a

1 2

1

n

n i n

i

s a a a a

lim

nn

s s

1 2

1

orn nn

a a a s a s

7

The sum of a series is the limit of the sequence of partial sums.

◦ i.e., when we write , we mean that, by

adding sufficiently many terms of the series, we can get as close as we like to the number s.

Or: The total sum is the limit of the sequence of

partial sums

1

nn

a s

1 1

lim

n

n in

n i

a a

8

An important example of an infinite series is the geometric series

Each term is obtained from the preceding one by multiplying it by the common ratio r.

2 3 1

1

1

0

n

n

n

a ar ar ar ar

ar a

9

The geometric series

is convergent if |r | < 1, and its sum is

If |r | ≥ 1, the series is divergent.

1 2

1

nn

ar a ar ar

1

1

. .1

,1

11

n

n

st term

common ratie

oi

aar r

r

10

Determine whether the geometric series is convergent or divergent. If convergent, find its sum.

SOLUTIONS

0

2 1

1

51.

4

2. 3 5

kk

n n

n

https://mathfixation.files.wordpress.com/2012/06/examples-1-4.pdf

11

Express the number as a ratio of integers.

SOLUTIONS

3. 0.784

4. 6.254

https://mathfixation.files.wordpress.com/2012/06/examples-1-4.pdf

12

The harmonic series

is divergent. This is one of the most important of all diverging

series. It arises in connection with the overtones produced by a vibrating musical string.

It is not immediately apparent that it diverges, but will eventually become apparent.

1

1 1 1 11

2 3 4n n

13

If the series is convergent, then

NOTE: The converse of this theorem is not

necessarily true as shown in next theorem.

1

nn

a

lim 0

nn

a

14

If does not exist or if , then the series is divergent.

If then the series may converge or

diverge.

lim nn

a

lim 0nn

a

1

n

n

a

lim 0nn

a

1

n

n

a

15

Determine whether the series is convergent or divergent. If it is convergent, find its sum.

SOLUTIONS

1

5.1

k

k

k

https://mathfixation.files.wordpress.com/2012/06/examples-5-9.pdf

16

If Σan and Σbn are convergent series, then so are the series Σcan (where c is a constant), Σ(an + bn), and

Σ (an – bn), and

1 1

1 1 1

1 1 1

i.

ii.

iii.

n n

n n

n n n n

n n n

n n n n

n n n

ca c a

a b a b

a b a b

17

Determine whether the series is convergent or divergent. If it is convergent, find its sum.

SOLUTIONS

11

1

10

3 26.

4 5

57.

18.

n nn

k

k

k

k

https://mathfixation.files.wordpress.com/2012/06/examples-5-9.pdf

18

Determine whether the series is convergent or divergent by expressing sn as a telescoping sum. If is it convergent, find its sum.

SOLUTIONS

21

221

29.

4 3

1 110. cos cos

1

n

n

n n

n n

https://mathfixation.files.wordpress.com/2012/06/examples-10-and-practice.pdf

19

Determine whether the series is convergent or divergent. If it is convergent, find its sum.

SOLUTIONS

1

1

1

21

1 31.

2

2. 0.8 0.3

3.

n

nn

n n

n

k

k

e

k

https://mathfixation.files.wordpress.com/2012/06/examples-10-and-practice.pdf

20

Find the values of x for which the series converges. Find the sum of the series for those values of x.

SOLUTIONS

1

0

4. 4

35.

2

n

n

k

kk

x

x

https://mathfixation.files.wordpress.com/2012/06/examples-10-and-practice.pdf

http://www.math.psu.edu/dlittle/java/calculus/sequences.html

http://calculusapplets.com/series.html

http://youtu.be/6LfoMmCckFY

http://www.math.psu.edu/dlittle/java/calculus/sequences.htmlhttp://www.math.psu.edu/dlittle/java/calculus/sequences.htmlhttp://calculusapplets.com/series.htmlhttp://youtu.be/6LfoMmCckFY