Convergence of series of functions implies uniform convergence of series of functions

Question

So I have, for $x\in(0,\infty)$,

$$\sum\limits_{n=0}^\infty \frac{x}{(1+nx)(1+(n+1)x)}$$ Which is equal to $$\sum\limits_{n=0}^\infty \frac{1}{(1+nx)}-\frac{1}{(1+(n+1)x)}$$ Hence the partial sum formula is, for $k\in\mathbb{N}$, $$S_k(x)=1-\frac{1}{1+(k+1)x}$$ Hence $\lim\limits_{k\to\infty} S_k(x)=1$. Is this sufficient to conclude that the series converges uniformly on this interval?

Answer 1

Hint Uniform convergence on $(0,\infty )$ would mean that we can make

$|S_k(x) -1| = \frac{1}{1+(k+1)x}$

as small as we want by choosing $k$ independently of $x.$