Investigate whether the series $\sum_{n=1}^\infty \frac{1+x^2 \sin (5x)}{\sqrt{n^3}}$ converges uniformly on $[-4,2]$ or not.

Question

Investigate whether or not the series $\sum_{n=1}^\infty \frac{1+x^2 \sin (5x)}{\sqrt{n^3}}$ converges uniformly on $[-4,2]$.

Weierstrass M-Test. Let $(f_n)$ be a sequence of functions on $D \in \Bbb R$ to $\Bbb R$. Let $(M_n)$ be a sequence of positive real numbers such that $|f_n(x)|\le M_n$ for all $x \in D, n \in \Bbb N$. If the series $\sum_{n=1}^\infty M_n$ convergent, then $\sum_{n=1}^\infty f_n$ uniformly convergent on $D$.

Attempt: Notice that for all $x \in [-4,2]$, we have:

1. $x^2 \le 16$, which is achieved when $x=-4$.
2. $|\sin(5x)| \le 1$.

Therefore, for all $x \in [-4,2]$ and any $n \in \Bbb N$, we have \begin{align*} \left|\frac{1+x^2 \sin(5x)}{\sqrt{n^3}}\right| \le \frac{1+|x^2 \sin(5x)|}{\sqrt{n^3}}\le \frac{1+x^2}{\sqrt{n^3}} \le \frac{1+16}{\sqrt{n^3}} = \frac{17}{\sqrt{n^3}}. \end{align*} Hence, we get a sequence of positive real numbers $\left(\frac{17}{\sqrt{n^3}} \right)$ such that $\left|\frac{1+x^2 \sin(5x)}{\sqrt{n^3}}\right| \le \frac{17}{\sqrt{n^3}}$ for all $x \in [-4,2]$ and $n \in \Bbb N$. Now, the series $\sum_{n=1}^\infty \frac{17}{\sqrt{n^3}}$ is convergent by the $p$-Series Test. Thus, by the Weierstrass M-Test, we conclude that the series $\sum_{n=1}^\infty \frac{1+x^2 \sin (5x)}{\sqrt{n^3}}$ converges uniformly on $[-4,2]$. Q.E.D.

Does this approach correct? If not, how to get it correctly? Thanks in advanced for any comments and helps.

Answer 1

Your approach is right. I see no point to be deducted.