# Power series question.

Hi,

## Homework Statement

I am asked to prove that if the power series Ʃ(1,infinity) a_n(x-x0)^n converges at a point d, then it converges for every x that satisfies |x-x0|<|d-x0|.

## The Attempt at a Solution

Obviously |d-x0|<r, where r denotes the radius of convergence. Should I strive to demonstrate that every such x would also satisfy |x-x0|<r? Is that the way to go about proving the above proposition? There is a general rule that if a power series converges at a point a, then it converges at every point x satisfying |x|<|a|. Should I therefore strive to show that |x|<|d|, or am I actually requested to prove the general rule? I'd appreciate some guidance.

haruspex
Homework Helper
Gold Member
2020 Award
It seems to me you are being asked to prove that there is such a thing as radius of convergence, so better not to assume it. But it is hard to be sure.

What would you suggest then?

haruspex
Homework Helper
Gold Member
2020 Award
If in doubt, go back to first principles. Given ε > 0, there exists... You should be able to use this criterion wrt x-x0 to show satisfied for some x closer x0.

So what keeps me from letting epsilon be the radius itself?

haruspex