Using Weierstrass M-test determine the set on which the series `\sum 1/n^{3/2} (x/{1+x})^n` converges absolutely uniformly.

Solution:

To determine the set on which the series `\sum 1/n^{3/2} (x/{1+x})^n` converges absolutely uniformly, we can utilize the Weierstrass M-test. This test allows us to establish uniform convergence by comparing the series to a convergent series with known bounds.

Let's analyze the given series: `\sum 1/n^{3/2} (x/{1+x})^n`.

First, consider the sequence of functions `f_n(x) = 1/n^{3/2} (x/{1+x})^n`. To apply the M-test, we need to find a convergent series `\sum M_n` such that `|f_n(x)| ≤ M_n` for all x.

For the given series, note that `x/{1+x}` is always less than or equal to 1 on the interval [0,1]. Therefore, we can simplify the inequality as follows:

`|f_n(x)| = 1/n^{3/2} (x/{1+x})^n ≤ 1/n^{3/2}`

Now, let's consider the series `\sum 1/n^{3/2}`. This series is a convergent p-series with p = 3/2, which means it converges. Therefore, we can choose `M_n = 1/n^{3/2}` as our convergent series.

Now, we need to show that `|f_n(x)| ≤ M_n = 1/n^{3/2}` for all x in the interval [0,1]. Since we already established that `x/{1+x} ≤ 1`, we can rewrite the inequality as:

`1/n^{3/2} (x/{1+x})^n ≤ 1/n^{3/2}`

This inequality holds for all x in the interval [0,1] and for all n. Hence, the series `\sum 1/n^{3/2} (x/{1+x})^n` satisfies the conditions of the Weierstrass M-test.

By the M-test, if a series of functions `\sum f_n(x)` converges absolutely and uniformly on a set S, then it also converges uniformly on S. Since the series `\sum 1/n^{3/2} (x/{1+x})^n` is absolutely convergent and satisfies the conditions of the M-test on the interval [0,1], we can conclude that the series converges absolutely and uniformly on the interval [0,1].

In summary, the series `\sum 1/n^{3/2} (x/{1+x})^n` converges absolutely and uniformly on the interval [0,1].