Equivalence of Two Infinite Series

[Bashyboy]

I am having a difficult time seeing how $\sum_{n=0}^{\infty} ((-1)^n + 1)x^n$ is equivalent to $2\sum_{n=0}^{\infty} x^{2n}$

 

[Simon Bridge]

What is the difficulty?

take a look at $((-1)^n +1)$, if n is even, what is it? If n is odd, what is it?
or just write out the first 5 or so non-zero terms of each and compare.

 

[Bashyboy]

Oh, I see. So, any odd power would give a trivial answer, and we would disregard those?

 

[Simon Bridge]

Oh, I see. So, any odd power would give a trivial answer, and we would disregard those?

It is not "trivial" it is zero. Zero terms do not change the sum so it does not matter if you regard them them or not.

Sometimes the notation can hide stuff if you are not used to it - when stuck, try writing out a bunch of terms.

 

[Bashyboy]

Thank you for your help.