# Convergence of alternating series

1. Mar 24, 2009

### vikkisut88

1. The problem statement, all variables and given/known data
Let s be the sum of the alternating series $$\sum$$(from n=1 to $$\infty$$)(-1)n+1an with n-th partial sum sn. Show that |s - sn| $$\leq$$an+1

2. Relevant equations
I know about Cauchy sequences, the Ratio test, the Root test

3. The attempt at a solution
I really have no idea where i'm meant to start with this question. I didn't know whether I thne have to take more partial sums i.e. partial sums of the n-th partial sum and work with it from there, or something else. I would be really grateful if someone could help me out.

2. Mar 24, 2009

### CompuChip

I don't think your statement is true as you gave it.
For example, let
$$\begin{cases} a_0 & {} = 1 \\ a_1 & {} = 0 \\ a_n & {} = 2^{-n} \qquad (n > 1) \end{cases}$$

Then s = 3/2, |s - s0| = |5/6 - 1| = |-1/6| = 1/6 which is not smaller than a1 = 0.

I had s = 5/6, I think it should be 3/2. Doesn't affect my argument though.[/edit]

Last edited: Mar 24, 2009
3. Mar 24, 2009

### vikkisut88

sorry i don't really understand that - how did you work out that s was 5/6? And did you just choose random values for a0, a1 and an? I have rechecked my homework question and that is exactly what it said!

4. Mar 24, 2009

### Dick

CompuChip just gave you a counterexample. To prove your conclusion, you also need to assume that a_n is a nonnegative sequence decreasing to zero monotonically. That might be what you are missing.

5. Mar 24, 2009

### vikkisut88

okay, but i still don't understand how i'm meant to show the result, sorry. This question has got me completely flummoxed.

6. Mar 24, 2009

### Dick

Start by looking at the difference of two partial sums |s_m-s_n| for m<n. Try to show that's less than or equal to a_m+1. Write down the terms making up the difference and regroup them. This would show the sequence is Cauchy, right?