Stuck At the Last Step Simplifying A Series

[ross1219]

Homework Statement

Find a power series for the function, centered at c, and determine the interval of convergence.

Homework Equations

g(x) = 2 / (1-x^2) , c=0

The Attempt at a Solution

2 / (1+x)(1-x) = 1/(1+x) + 1/(1-x)

1/(1-(-x)) => Sum [(-1)^n (x^n)], n=0 to infinity. Converges when abs(x) < 1, (-1,1)
1/(1-x) => Sum x^n, n=0 to infinity. Converges when abs(x) < 1, (-1,1)

So, 2/(1-x^2) = Sum[(-1)^n + 1]x^n, n=0 to infinity.

OK, here's the part that is probably SO simple, but I'm just not seeing it. The book shows the series above, Sum[(-1)^n + 1]x^n, n=0 to infinity is then equal to:

Sum 2x^(2n), n=0 to infinity.

How do you do that last simplification? Sorry for such a simple question, but I'm stuck. :)

Also, the book has the interval of convergence for the series as abs(x^2)<1 or (1,1), which I'm pretty sure is just a typo. It should be (-1,1), right?

 

[scurty]

Try writing out the first few terms of \displaystyle\sum_{n=0}^{\infty}((-1)^n+1)x^n and see what kind of pattern emerges. Then convert it back into summation notation!

Also, I believe that was a typo that your book had. (1,1) technically isn't even a number, never mind a range of numbers.

 

[ross1219]

Thank you so much scurty! I should have tried that, but thought I was missing something basic algebraically. I truly appreciate your help!

 

[scurty]

No problem! I always write series out to see if I find patterns because I'm not really good at visualizing it otherwise.

 

[ross1219]

Yes, I learned at least two good lessons here. Write out some terms of a series if necessary, and this forum rocks! Thanks again. :)