Strategy for Series

Consider a series in the form . To determine whether the series converges or diverges, one of these cases (the most appropriate one) can be used. No need to memorize all these.

1. If is apparent, then divergence test can be used. Otherwise look out for another way.
2. If consists of a constant raised to a power of , geometric series can be used.
3. If consists of raised to a constant power, p-series can be used.
4. If consists of , alternating series test can be used.
5. If consists of raised to a power of , root test would be suitable.
6. If includes , ratio test must be used.
7. If is a fraction, and consists of in both the denominator and the numerator, then direct comparison test or limit comparison test can be used. Consider the dominating parts to choose the .
8. If is a positive and decreasing function, and is easy to evaluate, then integral test can be used.

Secret note on inequalities

For any and , as tends to , the below inequality holds:

These inequalities can be used to find corresponding to some to be used for direct comparison test or limit comparison test. The list is found on a video by blackpenredpen on YouTube.

Also: