# if $\sum@\slimits@@@_{k=1}^{\infty}a_{k}$ converges then $a_{k}\to 0$

###### 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. 1.

The harmonic series $\sum_{k=1}^{\infty}1/k$ shows that the implication can not be reversed.

2. 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 $N\geq 1$ such that

 $|\sum_{k=1}^{M}a_{k}-S|<\frac{\varepsilon}{2}$

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

 $\displaystyle|a_{j+1}|$ $\displaystyle=$ $\displaystyle|\sum_{k=1}^{j+1}a_{k}-\sum_{k=1}^{j}a_{k}|$ $\displaystyle\leq$ $\displaystyle|\sum_{k=1}^{j+1}a_{k}-S|+|S-\sum_{k=1}^{j}a_{k}|$ $\displaystyle<$ $\displaystyle\varepsilon,$

and the claim follows. ∎

Title if $\sum@\slimits@@@_{k=1}^{\infty}a_{k}$ converges then $a_{k}\to 0$ Ifsumk1inftyAkConvergesThenAkto0 2013-03-22 15:00:38 2013-03-22 15:00:38 matte (1858) matte (1858) 13 matte (1858) Theorem msc 40-00 necessary condition of convergence DeterminingSeriesConvergence CompleteUltrametricField ConvergenceConditionOfInfiniteProduct LambertSeries AbsoluteConvergenceOfIntegralAndBoundednessOfDerivative ConvergentSeriesWhereNotOnlyA_nButAlsoNa_nTendsTo0