Thread: Abel's Theorem for power series

1. Abel's Theorem for power series

Hi everyone,

I'm reading my analysis textbook and trying to prove Abel's Theorem but I don't really get it. Any help would be very much appreciated.

Suppose that R in (0, ∞) is the radius convergence for the power series ∑(k=0 to inf) a_k(x-a)^k.

a. if ∑(k=0 to ∞) (a_k)*R^k converges, then ∑(k=0 to inf) a_k(x-a)^k converges uniformly on [a-R+ ε1, a+R] for any ε1>0.
b. if ∑(k=0 to ∞) (a_k)* (-R)^k converges, then ∑(k=0 to inf) a_k(x-a)^k converges uniformly on [a-R, a+R-ε2] for any ε2 >0.

Thank you very much!

2. Originally Posted by EmmWalfer
Hi everyone,

I'm reading my analysis textbook and trying to prove Abel's Theorem but I don't really get it. Any help would be very much appreciated.

Suppose that R in (0, ∞) is the radius convergence for the power series ∑(k=0 to inf) a_k(x-a)^k.

a. if ∑(k=0 to ∞) (a_k)*R^k converges, then ∑(k=0 to inf) a_k(x-a)^k converges uniformly on [a-R+ ε1, a+R] for any ε1>0.
b. if ∑(k=0 to ∞) (a_k)* (-R)^k converges, then ∑(k=0 to inf) a_k(x-a)^k converges uniformly on [a-R, a+R-ε2] for any ε2 >0.

Thank you very much!
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