Analysis of Infinite Series and Finding their Sums

The first step in analyzing an infinite series is to decide whether it even converges or not.

If the infinite converges then one can attempt to find its sum (limit).

A very simple check for divergence that often (but not always) works is whether the terms themselves tend to 0. If the terms do not go to 0, then the series diverges.

To test for convergence you can try the plethora of convergence tests: integral tests, ratio

tests, root tests, comparison tests, etc.

Most of the theory concerning infinite series is to decide whether the series diverges or converges.
Error estimates are available for both positive term and alternating term infinite series.

An infinite series is, by its very definition, always the limit of sequence of its partial sums.

In general, when dealing with arbitrary series, there is no uniform strategy/approach/procedure
to find the limit of the sequence of partial sums.

The integral tests and the comparison tests are both very useful for positive term series. For finding the sums themselves, there are actually no good overall strategies.

∞

The sum of a convergent infinite geometric series, ∑ arⁿ is S = a/(1 - r)

n=0

The sum of a convergent telescoping series is found by computing the partial sums until all

remaining terms cancel. The sequence of partial sums then yields the sum of the series.

The computation of a series of infinite products often involves trickery and ingenuity rather than
theory and depends highly on a case-by-case basis. Usually there is some kind of "product

telescoping" involved, i.e. when you compute the partial products.

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