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Homedetermining series convergence
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determining series convergence
Consider a series $\Sigma a_{n}$. To determine whether $\Sigma a_{n}$ converges or diverges, several tests are available. There is no precise rule indicating which type of test to use with a given series. The more obvious approaches are collected below.

If the series is an alternating series, then the alternating series test may be used.

Abel’s test for convergence can be used when terms in $\Sigma a_{n}$ can be obained as the product of terms of a convergent series with terms of a monotonic convergent sequence.
The root test and the ratio test are direct applications of the comparison test to the geometric series with terms $(a_{n})^{{1/n}}$ and $\frac{a_{{n+1}}}{a_{n}}$, respectively.
For a paper about tests for convergence, please see this article.
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Comments
Lacks first condition to convergence
If I'm not mistaken, this lacks the first condition to the convergence of a serie: If the serie is convergent then the limit of its main term (the sequence that constitutes the serie) is zero.

Nuno Morgadinho
Undergraduate Computer Science Student
Ã‰vora University