Abel's test

In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters.

Suppose the following statements are true:

Then ${\displaystyle \sum a_{n}b_{n}}$ is also convergent.

It is important to understand that this test is mainly pertinent and useful in the context of non absolutely convergent series ${\displaystyle \sum a_{n}}$. For absolutely convergent series, this theorem, albeit true, is almost evident.

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