if $\sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{k=1}^{\mathrm{\infty }}{a}_k$ converges then $a_k\to 0$

Theorem 1.

Suppose $a_{\mathrm{1}}\mathrm{,}{a}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }$ is a sequence of real or complex numbers. If the series

$$\sum _{k=1}^{\mathrm{\infty }}{a}_k$$

converges, then ${\lim }_{k\mathrm{\to }\mathrm{\infty }}\mathit{}{a}_k\mathrm{=}\mathrm{0}$.

Remarks

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

2. This result can be used as a first test for convergence of a series ${\sum }_{k=1}^{\mathrm{\infty }}{a}_k$. If $a_k$ does not converge to $0$, then ${\sum }_{k=1}^{\mathrm{\infty }}{a}_k$ does not converge either.

Proof.

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

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

$|a_{j+1}|$ $=$ $|{\displaystyle \sum _{k=1}^{j+1}}{a}_k-{\displaystyle \sum _{k=1}^j}{a}_k|$

$\le$ $|{\displaystyle \sum _{k=1}^{j+1}}{a}_k-S|+|S-{\displaystyle \sum _{k=1}^j}{a}_k|$

$\epsilon ,$

and the claim follows. ∎

Title	if $\sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{k=1}^{\mathrm{\infty }}{a}_k$ converges then $a_k\to 0$
Canonical name	Ifsumk1inftyAkConvergesThenAkto0
Date of creation	2013-03-22 15:00:38
Last modified on	2013-03-22 15:00:38
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	13
Author	matte (1858)
Entry type	Theorem
Classification	msc 40-00
Synonym	necessary condition of convergence
Related topic	DeterminingSeriesConvergence
Related topic	CompleteUltrametricField
Related topic	ConvergenceConditionOfInfiniteProduct
Related topic	LambertSeries
Related topic	AbsoluteConvergenceOfIntegralAndBoundednessOfDerivative
Related topic	ConvergentSeriesWhereNotOnlyA_nButAlsoNa_nTendsTo0