# Series

## Summable Sequence

The sequence is called summable if the sequence converges, where

In this case, is denoted by and is called the sum of the sequence

## Cauchy Criterion

The sequence is summable if and only if

Therefore, if is summable then

## Divergence of

Grouping the series as

we see that the sum of the terms in each ( ) is ; thus is unbounded. Therefore the series is not summable.

## Geometric Series

The infinite series is called the geometric series. If then each individual term does not converge to so the series diverges. If , then it follows from the two equations

that

or

Since , it follows that

# Convergence Criterion for Nonnegative Series

## Boundedness Criterion

A nonnegative series is convergent if and only if the set of partial sums is bounded.

## Comparison Test

1. Suppose that

Then if converges, so does

Proof. If

then

Since converges, is bounded. Therefore is bounded; consequently is convergent.

2. If , and , then converges if and only if converges.

Proof. Suppose that converges. Since , there exists some such that

Since the series converges, it follows that the series converges and this implies the convergence of the whole series which has only finitely many additional terms.

The converse follows at once since

## The Ratio Test

Let for all , and suppose that

(a) If then converges.

(b) If then the terms of do not approach so diverges.

Proof. (a) Suppose that . Fix with . The assumption implies that there exists some such that

i.e.,

Thus

Since converges, it follows that converges. Therefore the whole series converges.

(b) Suppose that . Fix with . Then there exists some such that

which means

This shows that the individual terms of do not approach , so is not summable.

## The Root Test

Let for all , and suppose that

(a) If then converges.

(b) If then the terms of do not approach so diverges.

Proof. (a) Suppose that . Fix with . The assumption

implies that there is some such that

i.e.,

Thus

Since the geometric series converges, it follows that converges. Therefore the whole series converges.

(b) Suppose that . Fix with . Then there exists some such that

which means

This shows that the individual terms of do not approach , so is not summable.

## Convergence of the Exponential Series

Let be any positive number. The ratio

converges to as approaches infinity. Hence it follows from the ratio test that the series converges for any positive number . Consequently we have

## Improper Integral

If the limit exists, it is denoted by , and called an improper integral.

• Find if .

• Show that does not exist.

• Suppose that for and that exists. Show that if for all , then exists.

The improper integral is defined as . If both improper integrals and exist, then the improper integral exists and equals .

Note that does not exist although the limit exists.

• Prove that if exists then exists and equals

There is another kind of improper integral in which the interval is bounded, but the function is unbounded. If is an unbounded function such that for any with is bounded on and the limit

exists, then the improper integral is said to exist and is equal to Similarly, if is an unbounded function such that for any with is bounded on and the limit

exists, then the improper integral is said to exist and is equal to

## Integral Test

Suppose that is positive and decreasing on , and that for all . Then converges if and only if the improper integral

Proof. The convergence of is equivalent to the convergence of the series

Since is decreasing we have

Therefore, if exists, then the series converges and so it follows from the comparison test that converges, and so converges.

If converges, the second half of the inequality shows that the series converges and so the improper integral exists.

## Exercise

Establish convergence or divergence by a comparison test

1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

Decide for or against convergence based on the integral test

11. 12. 13. 14.

15. 16. 17. 18.

Decide convergence and name your test

19. 20. 21. 22. 23.

24. 25. (test all ) 26. (test all )

27. 28. (test all )

29. (a) Show that

(b) Express as a Riemann sum and show that approaches

30. Compute the sum of the telescoping series

31. Compute the sum

by considering the Taylor expansion of

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