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.
