AP Calculus

Calculus Methods

29 Determining Series Convergence

This is not so much a method, as a list of a convenient order

in which to look at a series, when asked to determine its

convergence. For more details on each of the below

processes, please see lessons 49 (telescoping and geometric),

these really require a method, since each method would be

approximately one step long.

1) Perform the nth term test: if the nth term does not go to zero,

the series does not converge. If the nth term goes to zero, the

series may converge or it may not.

2) If the series is an alternating series, perform the nth term test;

now the test is definitive. If you are working with an alternating

series and the nth term goes to zero, the series converges.

3) If it is a special series (geometric series or telescoping series),

work out the problem and write down the answer.

4) Perform the Root Test if every term is raised to a power

involving n.

5) Perform the Ratio Test if other powers are involved, or if

factorials are involved. Often times the Ratio Test is actually

easier than the Root Test, even when both can be used.

These last ones are very fluid in order. Some people like to do

them in one order, while others prefer a different order. It's up to

you!

6) Integral Test: don't forget to

a) Check to make sure the function is always decreasing. Try

finding the derivative to check this.

b) Check to make sure the function is always positive.

c) Check to make sure the function is continuous.

7) Limit Comparison Test

8) Direct Comparison Test

Good Luck!