Proving Convergence of Power Series for All x within Radius of Convergence

[peripatein]

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|.

Homework Equations

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]

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.

 

[peripatein]

What would you suggest then?

 

[haruspex]

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.

 

[peripatein]

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

 

[haruspex]

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

ε would be a bound in relation to the value to which the series converges, not in relation to x. Btw I should have said 'use the fact that this criterion is satisfied wrt d-x0 ...'.