# Infinite Series Strategy Sheet

Did I mention you should know the

Famous Series?

## Test for Divergence

Unless you immediately recognize a series (

*e.g.* it's one of the

Famous Series, or an obvious candidate for one of the

Convergence Tests) always start with this one. Why?

### At best, you could be done already!

### At worst, you'll have a great idea how to proceed:

then

you don't know what happens.

**Despair Not!** Your work was not in vain.

- Ask yourself:
**How** does go to zero?
- In the limit, does resemble terms in a famous series?
- Does the famous series have known convergence properties?
- If so, the problem series almost certainly has similar convergence properties. Set up a comparison with the famous series. The Limit Comparison Test is usually a good bet here, since you've already been looking at a similar limit.
- If not, go back and review Famous Series it's probably there.

Did I mention you should know the Famous Series?

## Limit Comparison Test

If you don't find an easy match to a

Famous Series, The

*Test for Divergence* will almost always provide you with a

Famous Series to use with the

*Limit Comparison Test.* Set up the ratio between individual terms of the unknown series and the

Famous Series and find the limit,

*L* . If 0 <

*L* < ∞ then the two series behave the same. If

*L* is 0 or ∞ with any luck, your

Famous Series "wins" the limit of the ratio in a useful way:

- Your unknown series converges if it is clearly smaller than a convergent Famous Series -- think about it.
- Your unknown series diverges if it is clearly larger than a divergent Famous Series -- think about it.

Did I mention you should know the Famous Series?

## Convergence Tests

### What are the various Convergence Tests?

### Which one should I use?

You have a number of

Convergence Tests available, and most series can be analyzed with more than one of them. The

Convergence Tests page has guidelines for diagnosing when a test is likely to work on a particular series.

-- DickFurnas - 17 Nov 2008