For an alternating series $\sum_{n=1}^{\infty}(-1)^n a_n$, where $a_n \geq 0$ is decreasing to $0$, we know $$S_{2n-1} \leq S \leq S_{2n}$$

where $S_n$ are the partial sums, then $$|S- S_n| \leq |S_{n+1} - S_n| = a_{n+1}$$

My question is how exactly do we get from the first (double) inequality to the second?


Have a look at this picture:

enter image description here

which shows the first five values of $s_n$, and imagine that $S$ is between $S_3$ and $S_4$. You can see that the formula $$ |S-S_n|\le|S_{n+1}-S_n| $$ is true because on the left side you have the distance between $S$ and $S_n$, which is less than (or equal to) the distance between $S_{n+1}$ and $S_n$. This claim descends from the monotonicity of $S_n$.

And $|S_{n+1}-S_n|$ is equal to $a_{n+1}$ for the definition of $S_n$.

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  • 1
    $\begingroup$ +$1$: This is exactly how I describe it to my students. $\endgroup$ – Clayton Jun 29 '18 at 15:32

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