Proof
(related to Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero)
- If the infinite series \(\sum_{k=0}^\infty x_k\) is convergent, then by Cauchy's criterion, for a given \(\epsilon > 0\), we can find an index \(N(\epsilon)\in\mathbb N\) such that \[\left|\sum_{k=m}^n x_k\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).\]
- In particular, for \(m=n\) we have \[|x_n| < \epsilon\quad\quad \text{for all}\quad n\ge N(\epsilon).\]
- Thus, it follows \[\lim_{n\to\infty} x_n=0.\]
- It remains to be shown that we cannot conclude from \(\lim_{n\to\infty} x_n=0\) that \(\sum_{k=0}^\infty x_k\) is convergent. It suffices to find at least one counter-example. This counter-example can be the harmonic series \[\sum_{n=1}^\infty \frac 1n,\]
which is divergent, but for which \(\lim_{n\to\infty}\frac 1n=0\).
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
