Alternating Series Test

Prove the "Alternating Series Test"; .ie suppose that (x_n) is a positive
decreasing sequence with lim (x_n)= 0. Show that the alternating series

∞
∑(-1)^n (x_n)
n=0

is convergent and the sum satisfies |s-s_k|≤ s_k

where s_k is the partial sum and s = lim (s_k).

Hint: Start by showing that (s_2k) and (s_2k+1) are both monotone.



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I showed that the odd partial sums are monotone increasing and the even partial sums are monotone decreasing, but I couldn't find the rest.

If you help me, I will be really happy.

Selin

Hello,

Quote:

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Originally Posted by selinunan

Prove the "Alternating Series Test"; .ie suppose that (x_n) is a positive
decreasing sequence with lim (x_n)= 0. Show that the alternating series

∞
∑(-1)^n (x_n)
n=0

is convergent and the sum satisfies |s-s_k|≤ s_k

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Don't you mean $\displaystyle |s-s_k|\leq x_{k+1}$ ?

Quote:

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I showed that the odd partial sums are monotone increasing and the even partial sums are monotone decreasing, but I couldn't find the rest.

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Can you show that $\displaystyle (s_{2k})$ is bounded from below and that $\displaystyle (s_{2k+1})$ is bounded from above ? Hint (highlight to read) : * show that for all k, s_{2k+1}<s_{2k} *

In the question it is written as the way I wrote:

|s-s_k|≤ s_k

But I think it should have be written as you have said.

It can't be solved if it's |s-s_k|≤ s_k , can it?