We recall here the most important tests for convergence, which will be useful in the next sections. There exist other tests; the interested reader can see them in the Calculus Tutorial.

$\mathbb{C}$

- Suppose that there exists a convergent series such that, for a given natural number , . Then the series is convergent.
- Suppose that there exists a divergent series such that, for a given natural number , . then the series is divergent.

For , we have . By the integral test (consult the Calculus Tutorial), the series is convergent; thus the given series is convergent.

- If there exists a positive real number L such that , then the two series either are both convergent or are both divergent.
- If and is convergent, then is convergent.
- If and is divergent, then is divergent.

Take . Then:

Thus:

The series is a Riemann series with , thus it is convergent; therefore, by Thm 1.9, the given series is convergent.

- If , the series is convergent.
- If or , the series is divergent.
- If , the test gives no conclusion.

Thus:

As this limit is more than 1, the given series is divergent.

Thus:

By Thm 1.11, the given series diverges.

Noah Dana-Picard 2007-12-24