# MTH 251 Lecture Notes - Lecture 26: Alternating Series Test, Conditional Convergence, Direct Comparison Test

For unlimited access to Class Notes, a Class+ subscription is required.

MTH 251-Lecture 26-The ratio and root tests

Ratio Test Proof:

There is an N, so that if n,

For

Generally,

for

This series is geometric and converges, so,

converges, so

converges.

Example:

Does

converge or diverge?

Use the ratio test…

This is indeterminate, so, bring down the power,

So,

=

, so

is absolutely convergent by the ratio test.

Recall, however that it would be much quicker to show convergence by the comparison test.