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[parent] 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 $$ \sum_{k=1}^\infty a_k $$ converges, then $\lim_{k\to \infty} a_k = 0$ .

Remarks

  1. The harmonic series $\sum_{k=1}^\infty 1/k$ shows that the implication can not be reversed.
  2. This result can be used as a first test for convergence of a series $\sum_{k=1}^\infty a_k$ . If $a_k$ 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 $$ | \sum_{k=1}^M a_k -S | < \frac{\varepsilon}{2} $$ for all $M\ge N$ . For $j\ge N$ we then have \begin{eqnarray*} |a_{j+1}| &=& | \sum_{k=1}^{j+1} a_k -\sum_{k=1}^j a_k| \\ &\le & | \sum_{k=1}^{j+1} a_k -S | + |S - \sum_{k=1}^j a_k| \\ &<& \varepsilon, \end{eqnarray*}and the claim follows. $ \qedsymbol$




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See Also: determining series convergence, complete ultrametric field, convergence condition of infinite product, Lambert series, absolute convergence of integral and boundedness of derivative, convergent series where not only$~a_n$ but also $na_n$ tends to 0

Other names:  necessary condition of convergence
Keywords:  divergence test

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convergent series where not only$~a_n$ but also $na_n$ tends to 0 (Theorem) by pahio
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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.
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Classification:
AMS MSC40-00 (Sequences, series, summability :: General reference works )

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