If a real series converges then its terms tend to zero

Stub grade: B

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Routine for first years but also fairly important

Statement

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

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

Proof

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

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

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

By the addition of convergent sequences is a convergent sequence and the multiplication of convergent sequences is a convergent sequence we see:

• $\lim_{n\rightarrow\infty}(S_n-S_{n-1})\eq\ell-\ell\eq 0$
  • But $S_n-S_{n-1}\eq a_n$

So we see:

• $\lim_{n\rightarrow\infty}(a_n)\eq 0$

Grade: A*

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Make neater, had to leave before finishing. This is basically it though