How to show that the series of functions $\sum \frac{x^{2n}}{(n+x)^2}$ converges uniformly on $[0,1]$

Question

Here's the exercise I'm trying to answer: Determine whether or not the given series of functions converges uniformly on the indicated set. Justify your answers. a.) $\sum \frac{x^{2n}}{(n+x)^2}$ for $x\in[0,1]$

I know that it does converge uniformly on [0,1] (because the textbook tells you so), but I'm at a loss on how to justify it. I'm pretty sure I should show that the sequence of partial sums converges uniformly on [0,1] but I don't understand how I could do that... do I need to find an expression for the sequence of partial sums $(S_n)$? Or is there another way I can show that it converges uniformly?

Answer 1

Since $x \in [0,1]$, it follows

$$n+x\ge n \iff \dfrac{1}{n}\ge \dfrac{1}{n+x}$$

Now

$$x^{2n}\le 1^{2n}=1 \quad \forall n\in \mathbb{N}$$ Clearly we have: $$\dfrac{x^{2n}}{(n+x)^2}\le\dfrac{1}{n^2}$$ Finally apply comparison test.