Alternating series and the error bound

1. Statement. If $(a_n)$ is non-negative, decreasing and tends to $0$, then $S=\sum_{n\ge1}(-1)^{n+1}a_n$ converges, and $S$ lies between any two consecutive partial sums. Hence $|S-S_N|\le a_{N+1}$.

2. $a_n=\frac1{n^2}$ is positive, decreasing, and tends to $0$.

3. $|R_N|\le\frac1{(N+1)^2}<10^{-2}$ as soon as $N+1>10$, i.e. $N\ge10$.

Numerics. $S_9=0.82796$ and $S_{10}=0.81796$. The true sum $\frac{\pi^2}{12}=0.82247$ lies between them, within $10^{-2}$ of $S_{10}$.

Remark. Taking the *average* of two consecutive partial sums, $\frac{S_9+S_{10}}2=0.82296$, gives an error of only $5\cdot10^{-4}$: a cheap acceleration trick for alternating series.