Use the weierstrassm-test to show that each of the following series converges uniformly on the given domain:
on the domain [0,r], where 0≤r< 1.

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.

step-by-step explanation: