I would like to know what the name(s) are for different Methods for obtaining fast ways of approximating solutions to sums and/or products given a series or its partial sums.
I realise the time depends on the algorithm being used (implementation of method), how they are coded, the programming language, and resources available (plus probably more factors) affects the speed and time taken to evaluate to a desired accuracy.
Examples:
$S_1 = 1 + \frac{1}{2}+\frac{1}{3}+...=\sum_{n=0}^{\infty}\frac{1}{n}$ to 4 decimal places - can do this with an implementation of Method A that takes 10 seconds, or an implementation of Method B that takes 2 seconds.
$S_2=\sum_{n=0}^{\infty}\log_{10}n$ to 6 decimal places - can do this with an implementation of Method C that takes 30 seconds, or an implementation of Method D that takes 17 seconds.
$S_2=\sum_{n=0}^{\infty}n\sin(n)$ to 8 decimal places - can do this with an implementation of Method E that takes 60 seconds, or an implementation of Method F that takes 43 seconds.