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.