The comparison tests Theorem Suppose that and are series with

positive terms, then

(i) If is convergent and for all n, then is also convergent.

(ii) If is divergent and for all n, then is also

divergent. Ex. Determine whether converges.

Sol. So the series converges.

na nb

nanb n na b

nanb n na b

1

1

2 1nn

1 1

2 1 2n n

The limit comparison test Theorem Suppose that and are series with

positive terms. Suppose

Then

(i) when c is a finite number and c>0, then either both series

converge or both diverge.

(ii) when c=0, then the convergence of implies the

convergence of

(iii) when then the divergence of implies the

divergence of

na nb

lim .n

nn

ac

b

.nanb

,c nb.na

Example Ex. Determine whether the following series converges.

Sol. (1) diverge. choose then

(2) diverge. take then

(3) converge for p>1 and diverge for take

then

21

1(2)

ln ( 1)n n

2

51

2 3(1)

5n

n n

n

1

(3) sin p

n n

1/ 21/nb n lim / 2n nn

a b

1/nb n lim /n nna b

1/ pnb n1p

lim / pn n

na b

Question Ex. Determine whether the series

converges or diverges.

Sol.

1ln

1

( 0)n

n

a a

1 1ln ln ln

ln

1an n

n aa a e

n

diverge for 0 a e

converge for a e

Alternating series An alternating series is a series whose terms are alternati

vely positive and negative. For example,

The n-th term of an alternating series is of the form

where is a positive number.

1

1

1 1 1 ( 1)1

2 3 4

n

n n

nb

1( 1) or ( 1)n nn n n na b a b

The alternating series test Theorem If the alternating series

satisfies (i) for all n (ii)

Then the alternating series is convergent.

Ex. The alternating harmonic series

is convergent.

11 2 3 4 5 6

1

( 1) ( 0)nn n

n

b b b b b b b b

1

1

( 1)n

n n

1n nb b lim 0nn

b

Example Ex. Determine whether the following series converges.

Sol. (1) converge (2) converge

Question.

1 1 2

31 1

( 1) ( 1)(1) ( 0) (2)

1

n n

n n

n

n n

1

1

( 1)

4 1

n

n

n

n

Absolute convergence A series is called absolutely convergent if the series

of absolute values is convergent.

For example, the series is absolutely convergent

while the alternating harmonic series is not. A series is called conditionally convergent if it is

convergent but not absolutely convergent. Theorem. If a series is absolutely convergent, then it is

convergent.

na| |na

1

3/ 21

( 1)n

n n

na

Example Ex. Determine whether the following series is convergent.

Sol. (1) absolutely convergent

(2) conditionally convergent

21 1

sin ( 1)(1) (2)

ln(1 )

n

n n

n

n n

The ratio test The ratio test

(1) If then is absolutely convergent.

(2) If or then diverges.

(3) If the ratio test is inconclusive: that is, no

conclusion can be drawn about the convergence of

1lim 1,n

nn

aL

a

1n

n

a

1lim 1n

nn

aL

a

1lim n

nn

a

a

1n

n

a

1lim 1,n

nn

a

a

1n

n

a

Example Ex. Test the convergence of the series

Sol. (1) convergent

(2) convergent for divergent for

1 1

!(1) (2)

!

n n

nn n

a a n

n n

;a e a e1

11

( 1)! !/

( 1) (1 1/ )

n nn

n n nn

a a n a n a a

a n n n e

1 limn n nna e a a a

0

The root test The root test

(1) If then is absolutely convergent.

(2) If or then diverges.

(3) If the root test is inconclusive.

lim | | 1,nn

na L

1n

n

a

lim | | 1nn

na L

lim | |n

nn

a

1

nn

a

lim | | 1,nn

na

Example Ex. Test the convergence of the series

Sol.

convergent for divergent for

1

( 0)1

nn

na

an

1;a 0 1a

1lim lim

1

nn

nn n

na

aan

1 ( )1

(1 )n

n

na a n

n

Rearrangements If we rearrange the order of the term in a finite sum, then of

course the value of the sum remains unchanged. But this is not the case for an infinite series.

By a rearrangement of an infinite series we mean a series obtained by simply changing the order of the terms.

It turns out that: if is an absolutely convergent series with sum , then any rearrangement of has the same sum .

However, any conditionally convergent series can be rearranged to give a different sum.

s

na

nas

na

Example Ex. Consider the alternating harmonic series

Multiplying this series by we get

or

Adding these two series, we obtain

1/ 2,

1 1 1 1 11 ln 2.

2 3 4 5 6

1 1 1 1 1 1 1ln 2.

2 4 6 8 10 12 2

1 1 1 1 1 31 ln 2.

3 2 5 7 4 2

1 1 1 1 10 0 0 0 ln 2.

2 4 6 8 2

Strategy for testing series If we can see at a glance that then divergence

If a series is similar to a p-series, such as an algebraic form, or a form containing factorial, then use comparison test.

For an alternating series, use alternating series test.3

4( 1)

1n

n

na

n

1 10

2 1 2n

na

n

3

3 2 3/ 2

1 1~

3 4 2 3n

na

n n n

lim 0nn

a

Strategy for testing series If n-th powers appear in the series, use root test.

If f decreasing and positive, use integral test.

Sol. (1) diverge (2) converge (3) diverge (4) converge

( ),na f n

2nna ne

1

(ln )(ln ln )nan n n

ln

1 ( 1) ln 2 ! 1(1) tan (2) (3) (4)

( 2)! (ln )

n n

n

n n

n n nn

Power series A power series is a series of the form

where x is a variable and are constants called coefficients

of series. For each fixed x, the power series is a usual series. We can

test for convergence or divergence. A power series may converge for some values of x and

diverge for other values of x. So the sum of the series is a function

2 30 1 2 3

0

nn

n

c x c c x c x c x

2 30 1 2 3( )s x c c x c x c x

nc

Power series For example, the power series

converges to when

More generally, A series of the form

is called a power series in (x-a) or a power series centered

at a or a power series about a.

20 1 2

0

( ) ( ) ( )nn

n

c x a c c x a c x a

2 3

0

1n

n

x x x x

1( )

1s x

x

1 1.x

Example Ex. For what values of x is the power series

convergent? Sol. By ratio test,

the power series diverges for all and only converges

when x=0.

0

! n

n

n x

11 ( 1)!

lim lim lim( 1) | |!

nn

nn n nn

a n xn x

a n x

0,x

Homework 24 Section 11.4: 24, 31, 32, 42, 46

Section 11.5: 14, 34

Section 11.6: 5, 13, 23

Section 11.7: 7, 8, 10, 15, 36