Series with Positive terms: tests for

Convergence, Pt. 1The comparison test,

the limit comparison test, and the integral test.

Comparing series. . .Consider two series ,

with for all k.1 1

and k kk k

a b

kk ba 0

In this presentation, we will base all our series at 1, but similar results apply if they start at 0 or elsewhere.

Comparing series. . .Consider two series ,

with for all k.

How are these related in terms of convergence or divergence?

1 1

and k kk k

a b

kk ba 0

Note that:

1 1

1 2 1 2

1 2 3 1 2 3

1 2 3 4 1 2 3 4

And so on

a ba a b ba a a b b ba a a a b b b b

What does this tell us?

Comparing series. . .Consider two series ,

with for all k.

and k ka b

kk ba 0

Note that:

1 1

1 2 1 2

1 2 3 1 2 3

1 2 3 4 1 2 3 4

And so on

a ba a b ba a a b b ba a a a b b b b

What does this tell us?

Where does the fact that the terms are non-negative come in?

Series with positive terms. . .

1x

Since for all positive integers k. Then

0 kx

So the sequence of partial sums is . . . 1

n

n kk

s a

Non-decreasing Bounded above Geometric

1 2x x 1 2 3x x x 1 2 3 4x x x x

Back to our previous scenario. . . Consider two series ,

with for all k.1 1

and k kk k

a b

0 k ka b

Suppose that the series converges 1

kk

b

Suppose that the series converges

0 0 0

Note that for all positive integers ,

n n

k k kk k k

n

a b b

1k

k

b

Non-decreasing Bounded above Geometric

So the sequence of partial sums is . . . 1

n

n kk

s a

A variant of a familiar theorem

Suppose that the sequence is non-decreasing and bounded above by a number A. That is, . . .

ks

1 1 2 1 2 3a a a a a a A

Then the series converges to some value that is smaller than or equal to A.

1k

k

a

Theorem 3 on page 553 of OZ

Suppose that the series diverges

0 0

n n

k kk k

a b

1k

k

a

Non-decreasing Bounded below Unbounded

So the sequence of partial sums is . . . 1

n

n kk

s b

For all n we still have

This gives us. . . The Comparison Test: Suppose we have two series , with for all positive integers k.

If converges, so does , and

If diverges, so does .

1 1

and k kk k

a b

1k

k

b

1k

k

a

1k

k

b

1k

k

a

kk ba 0

A related test. . .

There is a test that is closely related to the comparison test, but is generally easier to apply. . . It is called the

Limit Comparison Test

This test is not in the book!

(One case of…) The Limit Comparison Test

Limit Comparison Test: Consider two series

with , each with positive terms.

If , then

are either both convergent or both divergent.

1 1

and k kk k

a b

n

n

ba

nlim

01 1

and k kk k

a b

Why does this work?

(Hand waving) Answer:

Because if

Then for “large” n, ak t bk. This means that “in the long run”

0 0

and have the same convergence behavior.n n

k kk k

a t b

tba

n n

n

lim

The Integral Test

y = a(x)

Now we add some enlightening pieces to our diagram….

Suppose that we have a sequence {ak} and we associate it with a continuous function y = a(x), as we did a few days ago. . .

The Integral Test

y = a(x)

Look at the graph. . .What do you see?

Suppose that we have a sequence {ak} and we associate it with a continuous function y = a(x), as we did a few days ago. . .

y = a(x)

The Integral Test

1a2a

3a4a

6a5a7a

If the integral

so does the series.

11

( ) ( )k

a x dx a k

So

converges

diverges

The Integral Test

Now look at this graph. . .What do you see?

y = a(x)

y = a(x)

The Integral Test

1a

2a3a

4a6a5a

7a

If the integral

so does the series.

12

( ) ( )k

a k a x dx

So

converges

diverges8a

Why 2?

The Integral TestThe Integral Test: Suppose for all x 1, the function a(x) is continuous,

positive, and decreasing. Consider the series and the integral .

If the integral converges, then so does the series.If the integral diverges, then so does the series.

1

( )k

a k

1( )a x dx

The Integral TestThe Integral Test: Suppose for all x 1, the function a(x) is continuous,

positive, and decreasing. Consider the series and the integral .

If the integral converges, then so does the series.If the integral diverges, then so does the series.

1

( )k

a k

1( )a x dx

Where do “positive and decreasing”

come in?