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Answer:
Step-by-step explanation:
Given a series , the Weierstrass M-test tell us that if we find a sequence of positive numbers
such that
in a certain domain D, and the series
converges, then the series
converges uniformly in the domain D.
So, our objective is to find the so called sequence . The main idea is to bound the sequence of functions
.
Now, notice that the values of z are always positive, so is always positive, so
for all values of z in
. Then,
because if we make the values of the denominator smaller, the whole fraction becomes larger.
Moreover, as z is in the interval [0,r], we have that and as consequence
. With this in addition to the previous bound we obtain
With this, our sequence is and the corresponding series is
, which is a geometric series with ratio less than 1, hence it is convergent.
Then, as consequence of Weierstrass M-test we have the uniform convergence of the series in the given domain.