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if $\sum_{k=1}^\infty a_k$ converges then $a_k\to 0$

(Theorem)

Theorem 1 Suppose $a_1,a_2, \ldots$ is a sequence of real or complex numbers. If the series

$\displaystyle \sum_{k=1}^\infty a_k$

converges, then $\lim_{k\to \infty} a_k = 0$ .

The harmonic series $\sum_{k=1}^\infty 1/k$ shows that the implication can not be reversed.
This result can be used as a first test for convergence of a series $\sum_{k=1}^\infty a_k$ . If does not converge to 0, then $\sum_{k=1}^\infty a_k$ does not converge either.

Proof. Let $S\in \mathbbmss{C}$ be the value of the sum, and let $\varepsilon>0$ be arbitrary. Then there exists an $N\ge 1$ such that

$\displaystyle \vert \sum_{k=1}^M a_k -S \vert < \frac{\varepsilon}{2}$

for all $M\ge N$ . For $j\ge N$ we then have

$\displaystyle \vert a_{j+1}\vert$	$\displaystyle =$	$\displaystyle \vert \sum_{k=1}^{j+1} a_k -\sum_{k=1}^j a_k\vert$
	$\displaystyle \le$	$\displaystyle \vert \sum_{k=1}^{j+1} a_k -S \vert + \vert S - \sum_{k=1}^j a_k\vert$
	$\displaystyle <$	$\displaystyle \varepsilon,$

and the claim follows. $\qedsymbol$

"if $\sum_{k=1}^\infty a_k$ converges then $a_k\to 0$ " is owned by matte.

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Other names:

necessary condition of convergence

Keywords:

divergence test

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Cross-references: sum, implication, harmonic series, converges, series, complex numbers, real, sequence
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This is version 10 of if $\sum_{k=1}^\infty a_k$ converges then $a_k\to 0$ , born on 2005-02-06, modified 2005-11-18.
Object id is 6717, canonical name is ThenA_kto0IfSum_k1inftyA_kConverges.
Accessed 3317 times total.

Classification:

40-00 (Sequences, series, summability :: General reference works )

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