When is it appropriate to neglect all terms after the first non-zero term of a Taylor expansion series?

Question

Suppose I am interested in the Taylor expansion series of a Cosine function at the neighbourhood of a=0. In computing the series from n=0 to n = infinity, when would it be appropriate to neglect all terms after the first non-zero term?

This is in regard to Bessel's equation

Answer 1

for $\cos(x)$, this is usually done when the argument $x$ is small compared with unity. Just how small depends on how accurate you would like your results.

The "quick and dirty" way to know if it's good enough, is to compare the error directly with evaluation of $\cos(x)$.