| Version 4 |
Version 3 |
| \begin{thm} Suppose $a_1,a_2, \ldots$ is a sequence of real or complex numbers. |
\begin{thm} Suppose $a_1,a_2, \ldots$ is a sequence of real or complex numbers. |
|
If the series
|
If the infinite sum
|
| $$ |
$$ |
| \sum_{k=1}^\infty a_k |
\sum_{k=1}^\infty a_k |
| $$ |
$$ |
| converges, then $\lim_{k\to \infty} a_k = 0$. |
converges, then $\lim_{k\to \infty} a_k = 0$. |
| \end{thm} |
\end{thm} |
|
|
| \begin{proof} Let $S\in \C$ be the value of the sum, and let $\varepsilon>0$ |
\begin{proof} Let $S\in \C$ be the value of the sum, and let $\varepsilon>0$ |
| be arbitrary. Then there exists an $N\ge 1$ such that |
be arbitrary. Then there exists an $N\ge 1$ such that |
| $$ |
$$ |
| | \sum_{k=1}^M a_k -S | < \frac{\varepsilon}{2} |
| \sum_{k=1}^M a_k -S | < \frac{\varepsilon}{2} |
| $$ |
$$ |
| for all $M\ge N$. For $j\ge N$ we then have |
for all $M\ge N$. For $j\ge N$ we then have |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| |a_{j+1}| &=& | \sum_{k=1}^{j+1} a_k -\sum_{k=1}^j a_k| \\ |
|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| \\ |
&\le & | \sum_{k=1}^{j+1} a_k -S | + |S - \sum_{k=1}^j a_k| \\ |
| &<& \varepsilon, |
&<& \varepsilon, |
| \end{eqnarray*} |
\end{eqnarray*} |
| and the claim follows. |
and the claim follows. |
| \end{proof} |
\end{proof} |
