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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.

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  • $\begingroup$ The test you refer to as intuition is the Cauchy Criterion or Cauchy Convergence Test: en.wikipedia.org/wiki/Cauchy's_convergence_test $\endgroup$ – Hayden Apr 7 '14 at 21:00
  • $\begingroup$ Depending on the series there are a few methods you could use. Do you have a formula for the $X_i$? $\endgroup$ – Antonio Vargas Apr 7 '14 at 21:24
  • $\begingroup$ Without any information about the $X_i$ other than that $\sum X_i$ converges, you cannot put a bound on $E_n = \left|\sum_{i=1}^\infty - \sum_{i=1}^n X_i\right| = \left|\sum_{i=n+1}^\infty X_i\right|$. The fact that the partial sums form a cauchy sequence doesn't help - you only know that for every $\epsilon$ there is an $N$, but you don't know how to actually find that $N$. OTOH, if you have a proof that the series is convergent, you might be able to extract an algorithmn for finding $N$ from it. If the proof is sufficiently constructive, that is... $\endgroup$ – fgp Apr 7 '14 at 21:30
  • $\begingroup$ i see. The series consists of independent random variables, where the conditions for Kolomogorov's one series theorem are satisfied, and thus used to prove a.s convergence. Page 46: books.google.no/… $\endgroup$ – Exatic Apr 7 '14 at 21:52
  • $\begingroup$ I will give it a try to extract the algorithm by myself, and eventually come back if i get stuck :) $\endgroup$ – Exatic Apr 7 '14 at 21:59
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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.

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