Alternating Series

Suppose $u_k>0$ . An alternating series is:

\sum_{k=1}^n (-1)^{k - 1} u_k = u_1 - u_2 + u_3 - u_4 + \cdots

Convergence Test

If $\forall k\; u_k>0$ , decreasing and $\lim_{n\to \infty} u_n = 0$ , then:

\sum_{k=1}^n (-1)^{k - 1} u_k \text{ is converging}

Proof

For odd-indexed elements:

s_{2m+3} \le s_{2m+1} \le s_1 = u_1

For even-indexed elements:

s_{2m+2} \ge s_{2m} \ge s_2 = u_1 - u_2

Combining these 2:

0 \le u_1 - u_2 \le s_2 \le s_{2m} \le s_{2m+1} \le s_1 = u_1

$s_{2m}$ is bounded above by $u_1$ and increasing. $s_{2m+1}$ is bounded below by $0$ and decreasing. So both converges.

\lim_{m\to \infty} (s_{2m+1} - s_{2m}) = \lim_{m\to \infty} u_{2m + 1} = 0

\implies \lim_{m\to \infty} s_{2m+1} = \lim_{m\to \infty} s_{2m} = s

Both converges to the same number.

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October 19, 2026 1 min read

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