Convergence of alternating series

[vikkisut88]

Homework Statement

Let s be the sum of the alternating series \sum(from n=1 to \infty)(-1)ⁿ⁺¹a_n with n-th partial sum s_n. Show that |s - s_n| \leqa_n+1

Homework Equations

I know about Cauchy sequences, the Ratio test, the Root test

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.

 

[CompuChip]

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

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

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

 

[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!

 

[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.

 

[vikkisut88]

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

 

[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?