# If a real series converges then its terms tend to zero

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Routine for first years but also fairly important

## Statement

Let [ilmath](a_n)_{n\in\mathbb{N} }\subseteq\mathbb{R} [/ilmath] be a real sequence, then:

• if the series corresponding to [ilmath]\sum_{n\eq 1}^\infty a_n[/ilmath] converges then [ilmath]\lim_{n\rightarrow\infty}\left(a_n\right)\eq 0[/ilmath]

## Proof

Here [ilmath]S_n:\eq\sum^n_{i\eq 1}a_n[/ilmath] denote the [ilmath]n[/ilmath]th partial sum of the series given by [ilmath](a_n)_{n\in\mathbb{N} } [/ilmath].

By definition of what it means for [ilmath]\sum^\infty_{n\eq 1}a_n [/ilmath] to converge know the limit: [ilmath]\lim_{n\rightarrow\infty}(S_n)[/ilmath] exists.

By UTLOC:1 we know that if [ilmath]\lim_{n\rightarrow\infty}(S_n)\eq\ell[/ilmath] (which it does) that [ilmath]\lim_{n\rightarrow\infty}(S_{n-1})\eq\ell[/ilmath] also

• [ilmath]\lim_{n\rightarrow\infty}(S_n-S_{n-1})\eq\ell-\ell\eq 0[/ilmath]
• But [ilmath]S_n-S_{n-1}\eq a_n[/ilmath]

So we see:

• [ilmath]\lim_{n\rightarrow\infty}(a_n)\eq 0[/ilmath]