- #1
applegatecz
- 14
- 0
Homework Statement
Find all x for which [tex]\sum[/tex] from k=1 to infinity (x^k - x^(k-1))(x^k+x^(k-1)) converges.
Homework Equations
I think the geometric series formula is relevant: [tex]\sum[/tex] k=N to infinity of x^k = 1/(1-x) for all |x|<1.
The Attempt at a Solution
I simplified the expression to x^2k - x^(2k-2). I can show that the first term converges (I think ... because it is the product of two convergent sequences?), and I understand logically why the second term converges, but not sure how to show rigorously. I also think that the series converges for all |x|<1, but again not sure how to construct the proof.