Applications of Convergence of a series in Algorithms

We were introduced to testing the convergence of a series & calculating the point of convergence in the first maths of college curriculum. I wish to explore its usage in computer algorithms.

What I mean is, the testing of convergence (using the different tests) gives an answer of yes | no. Such conclusions can save considerable amounts of computing time (as in algorithms where a no can mean, stop proceeding). I have seen the Taylor series being used in a few algorithms where the point of convergence is to be calculated.

Could anyone point out some real world algorithms where they play major role (e.g., optimization techniques, graphic algorithms, etc).

As Qiaochu Yuan stated in this post, "Mastering the use of Taylor series is already highly nontrivial - especially recognizing when the method is applicable". (Some examples of these non trivial places??)

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When teachers say that series are used in computing (special) functions "in real life", it's a bit of a lie; more often than not, due to the Taylor series only being useful near the point of expansion, and behaving poorly past that, usually algorithms that apply to wider argument ranges are used instead. – J. M. Aug 4 '11 at 6:39
That being said, some computations for functions that satisfy simple "doubling" relations (e.g. $\exp(2x)=\exp(x)^2$, $\cos(2x)=2\cos^2 x-1$) make use of Maclaurin series; briefly, keep halving the argument until "small", evaluate the series there, and then use the doubling relation for the function accordingly. However, there are almost always better approximations to use than Maclaurin series even for that purpose... – J. M. Aug 4 '11 at 6:42
Still another note: in computational practice, the "convergence tests" are almost never used; the usual criterion is "keep adding for as long as the terms are not too small", or |term/(partial sum)| >= (tolerance). – J. M. Aug 4 '11 at 6:44
Sometimes convergence is used to show that an algorithm is correct or behaves like it should, e.g. in machine learning and probabilistic algorithms. – chazisop Aug 4 '11 at 14:03