Arrange the numbers 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 so that the sum of the numbers on the vertical line is the same as the sum of the numbers in the horizontal line.

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.