First, let us recap what is Weierstrass M-test:

**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).

**Proof:**

Let . such that implies .

For ,

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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