Consider i have a series $\sum_{i=1}^{\infty} X_i $ which i know converges in $\mathbb{R}$,but don't know exactly where. I am trying numerically approximate to the convergence point but not sure when it's guaranteed that the error is smaller than a given number. I thought using the fact that the series as a sequence also is a Cauchy-sequence. So if
$$|\sum_{i=1}^{n} X_i - \sum_{i=1}^{k} X_i | < \epsilon , \forall k,n > N \in \mathbb{N}$$
for a given $\epsilon$, then the distance to convergence point must also be smaller than $\epsilon$ . This seems very intuitive, but I am not sure if it works.
You can use some theorems related to the convergence of series and estimination the sum of the series such as: Integralal's criteria, Dalambe's criteria, Alternating sign,...
In the paper of Larry Riddle "Approximating Sum of the Convergent Series"
you can see how the author approximates the sum of a convergent series:
http://ecademy.agnesscott.edu/~lriddle/apcalculus/approxSeries.pdf
I hope that the above reference can satisfy you.