First, let us recap what is Weierstrass M-test:
Let be a sequence of real (or complex)-valued functions defined on a set A, and let be a sequence satisfying
, and also .
Then, converges uniformly on A (to a function f).
Let . such that implies .
Thus, converges uniformly.
Application to prove Abel’s Theorem (Special Case):
Consider the special case of Abel’s Theorem where all the coefficients are of the same sign (e.g. all positive or all negative).
Then, for ,
Then by Weierstrass M-test, converges uniformly on [0,1] and thus .
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