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sandra1
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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.
Thank you very much!
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 ∞) 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 ∞) a_k(x-a)^k converges uniformly on [a-R, a+R-ε2] for any ε2 >0.
I know this has to do with the Weierstrass M-Test but I'm confused about the [a-R+ε1, a+R] and [a-R+ε2, a+R] parts:
So since R is radius of convergence , for |x-a|<R the series ∑(k=0 to ∞) a_k(x-a)^k converges, and so:
|a_k(x-a)^k| < (a_k)* (R)^k
and ∑(k=0 to ∞) (a_k)*R^k converges, so by the Weierstrass M-test ∑(k=0 to ∞) a_k(x-a)^k converges uniformly.
Could you tell me what do I need to do with connect with the [a-R+ε1, a+R] and [a-R+ε2, a+R]?
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.
Thank you very much!
Homework Statement
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 ∞) 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 ∞) a_k(x-a)^k converges uniformly on [a-R, a+R-ε2] for any ε2 >0.
Homework Equations
The Attempt at a Solution
I know this has to do with the Weierstrass M-Test but I'm confused about the [a-R+ε1, a+R] and [a-R+ε2, a+R] parts:
So since R is radius of convergence , for |x-a|<R the series ∑(k=0 to ∞) a_k(x-a)^k converges, and so:
|a_k(x-a)^k| < (a_k)* (R)^k
and ∑(k=0 to ∞) (a_k)*R^k converges, so by the Weierstrass M-test ∑(k=0 to ∞) a_k(x-a)^k converges uniformly.
Could you tell me what do I need to do with connect with the [a-R+ε1, a+R] and [a-R+ε2, a+R]?