# Thread: Proof of alternating series test

1. ## Proof of alternating series test

Hi,

I'm having trouble understanding the first line of Knapp's proof of the alternating series test. The Theorem states that

If for each $\displaystyle x$ in a nonempty set S, $\displaystyle \{ a_n(x)\}_{n \geq 1}$ is a monotone decreasing sequence of nonnegative real numbers such that $\displaystyle \lim_n a_n(x) = 0$ uniformly in $\displaystyle x$, then $\displaystyle \sum^{\infty}_{n=1}(-1)^n a_n(x)$ converges uniformly.

His opening line in the proof is

The hypotheses are such that $\displaystyle \left| \sum^{n}_{k=m}(-1)^k a_k(x) \right| \leq \sup_x |a_m(x)|$ whenever $\displaystyle n \geq m$...

It's the above line I'm totally confused by. Can someone help explain to me how the hypothesis implies the above?

Thanks

2. I won't give the complete details but the idea is the following. We have
$\displaystyle 0\leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}\leq a_m(x)$ and $\displaystyle à\leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}-a_{m+3}(x)\leq a_m(x)-a_{m+3}(x)\leq a_m(x).$

3. Originally Posted by girdav
I won't give the complete details but the idea is the following. We have
$\displaystyle 0\leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}\leq a_m(x)$ and $\displaystyle a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}-a_{m+3}(x)\leq a_m(x)-a_{m+3}(x)\leq 0.$
If $\displaystyle a_n(x)$ is nonnegative and monotone decreasing, how is $\displaystyle a_m(x)-a_{m+3}(x)\leq 0$?

4. Sorry, the second inequality is wrong, I meant $\displaystyle a_m(x)$ instead of 0. I will correct it.

5. Ok, I think I see it now; I'll work on writing the argument out in full. Thanks