$\sum \frac{x}{n^{a}(1+nx^{2})}$ converge uniformly in $\mathbb{R}$.

Question

Let $a>1/2.$ Determine if a series converge uniformly in $\mathbb{R}:$

$$\sum_{n=1}^{\infty} \frac{x}{n^{a}(1+nx^{2})}$$

If $x>0$ ... it is easy to prove that

$$\left |\frac{x}{n^{a}(1+nx^{2})} \right |\leq \frac{1}{2n^{a+\frac{1}{2}}} $$

is then only apply (Test- Weiersstras )... However, if $x < 0$ have difficulties. Some help?

Answer 1

Edit: after the OP's edit, only the second point remains relevant.

• You should find an upper bound of $$ \frac{1}{2n^{a+\frac{1}{2}}} $$ not $\frac{1}{2n^{a+1}}$. (Which makes "sense", given the assumption. Otherwise, why bother with $a>\frac{1}{2}$ specifically?) Namely, you want to find the maximum of $f\colon [0,\infty) \to \mathbb{R}$ defined by $f(x) = \frac{x}{1+nx^2}$, and this maximum occurs at $\frac{1}{\sqrt{n}}$.

• The LHS you are trying to upper bound is even (in $x$), so the argument automatically applies to $x< 0$ as well.