The following proposition shows that the convergence of terms in the sequence is a necessary condition for the series itself to converge.

Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero

A necessary, but not a sufficient1 condition for an infinite series $\sum_{k=0}^\infty x_k$ for being convergent is that the sequence of the summands \((x_n)_{n\in\mathbb N}\) converges to zero, formally $$\sum_{k=0}^\infty x_k\text{ convergent}\Longrightarrow\lim_{n\to\infty} x_n=0,$$ $$\lim_{n\to\infty} x_n= 0\not\Longrightarrow\sum_{k=0}^\infty x_k\text{ convergent}$$

$$\lim_{n\to\infty} x_n\neq 0\Longrightarrow\sum_{k=0}^\infty x_k\text{ divergent}$$

Proofs: 1

Proofs: 1 2 3

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983


  1. Please note that by the rule of contraposition, this statement can be used to prove that an infinite series does not converge if the series of its summands does not converge to zero, i.e.