Alternating_Series_Test

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Definition

Let $(a_n)$ be a sequence of positive numbers with $a_1 \geq a_2 \geq a_3 \geq \cdots$ and $\lim a_n = 0$. Then the alternating series $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$ converges.

Proof

Consider the partial sums $s_n = \sum_{k=1}^{n} (-1)^{k+1} a_k$.

The even partial sums satisfy $s_{2n} = (a_1 - a_2) + (a_3 - a_4) + \cdots + (a_{2n-1} - a_{2n})$. Since $a_k \geq a_{k+1}$, this is increasing.

Also $s_{2n} = a_1 - (a_2 - a_3) - \cdots - (a_{2n-2} - a_{2n-1}) - a_{2n} \leq a_1$, so $(s_{2n})$ is bounded.

By the Monotone_Convergence_Theorem, $(s_{2n}) \to s$ for some $s$.

Since $s_{2n+1} = s_{2n} + a_{2n+1}$ and $a_{2n+1} \to 0$, we have $s_{2n+1} \to s$ as well. Thus $s_n \to s$.