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The Cauchy Criterion for a series can be defined:

Theorem: A series $\sum_{k=1}^{\infty}a_k$ converges iff for all $\epsilon>0$ there is an $N\in \mathbb{N}$ so that for all $n\ge m \gt N$ we have $|\sum_{k=m+1}^{n} a_k|< \epsilon$.

My question is why does the sum start at m+1 and not at m, shouldn't the difference between the mth and nth term include the mth term?

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  • $\begingroup$ Yes,this should be start from $m$, but one might make $\sum_{k=m+1}^ma_k=0$. $\endgroup$ – Tony Ma Apr 7 '18 at 13:02
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$S_n=\sum_{k=1}^{n}a_k$ , $S_m=\sum_{k=1}^{m}a_k$ . So the difference is $\sum_{k=m+1}^{n}a_k$.

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  • $\begingroup$ Thanks, this wasn't immediately obvious to me. Is there a general rule for how the index of summation changes with operations on summations? $\endgroup$ – AinJalut Apr 7 '18 at 13:50
  • $\begingroup$ Not really. But the way, if you are confused with the sigma you can try to write the sum as $a_1+a_2+...+a_n$, sometimes it helps. $\endgroup$ – Mark Apr 7 '18 at 14:00
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Let $n\geq m>N$ an consider the partials sums $S_n$ and $S_m$ so, by Cauchy criterion we have $$ |S_n-S_m|=|\sum_{k=m+1}^na_k| $$

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