Decomposing a Series Using Riemann's Rearrangement Theorem

[the_kid]

Homework Statement

I'm trying to compute the sum of the following series:

S=1+\frac{1}{4}-\frac{1}{16}-\frac{1}{64}+\frac{1}{256}

Homework Equations

The Attempt at a Solution

I'm not really sure how to begin this one. I know it probably involves Riemann's Rearrangement Theorem since this series is absolutely convergent.

 

[Ray Vickson]

Homework Statement

I'm trying to compute the sum of the following series:

S=1+\frac{1}{4}-\frac{1}{16}-\frac{1}{64}+\frac{1}{256}

Homework Equations

The Attempt at a Solution

I'm not really sure how to begin this one. I know it probably involves Riemann's Rearrangement Theorem since this series is absolutely convergent.

Are the successive signs really ++--+? What happens after that? Are you sure you copied the question correctly?

RGV

 

[the_kid]

My apologies; I should have been clearer in my original post.

The signs are ++-- ++-- ...

 

[the_kid]

Any help?

 

[wjv4]

Okay, so I think that if you think about it as the sum of two sums, that will help...

think of the first sum as the sum of every odd indexed term, and the second sum as the sum of every even indexed term.

S₁ = 1-1/16+1/256...
S₂ = 1/4-1/64+1/(16*64)...

thus, the first sum will be

S₁ = \sum(-1/16)ⁿ from n = 0 to ∞.

and the second, I'll let you figure out. but I think that this should help. (note, the second one needs a constant out front.

hope this helps!