Terms in Convergent Series Converge to Zero

Theorem

Let $\sequence {a_n}$ be a sequence in any of the standard number fields $\Q$, $\R$, or $\C$.

Suppose that the series $\ds \sum_{n \mathop = 1}^\infty a_n$ converges in any of the standard number fields $\Q$, $\R$, or $\C$.

Then:

$\ds \lim_{n \mathop \to \infty} a_n = 0$

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.

Let $\sequence {x_j}_{j \mathop \in \N}$ be a sequence such that:

$\ds \sum_{j \mathop = 1}^\infty x_j$ converges.

Then:

$x_j \to {\mathbf 0}_X$ in $\struct {X, \norm {\, \cdot \,} }$ as $j \to \infty$.

Let $\ds s = \sum_{n \mathop = 1}^\infty a_n$.

Then $\ds s_N = \sum_{n \mathop = 1}^N a_n \to s$ as $N \to \infty$.

Also, $s_{N - 1} \to s$ as $N \to \infty$.

Thus:

$\ds a_N$

$=$

$\ds \paren {a_1 + a_2 + \cdots + a_{N - 1} + a_N} - \paren {a_1 + a_2 + \cdots + a_{N - 1} }$

$\ds $

$=$

$\ds s_N - s_{N - 1}$

$\ds $

$\to$

$\ds s - s = 0 \text{ as } N \to \infty$

Hence the result.

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.3$. Series
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 6.9$
1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests: Theorem $1.1$