Uniform Convergence of $\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1][nx+1]}$

Question

What can be said about the uniform Convergence of $\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1][nx+1]}$ in the interval $[0,1]$?

The sequence inside the summation bracket doesn't seem to yield to root or ratio tests. The pointwise convergence itself seems doubtful. Should we use Cauchy-Criterion directly here? Maybe some comparison would be useful in this case? And what about uniform convergence? Any hints? Thanks beforehand.

Answer 1

Hint: $\displaystyle\frac1{\bigl((n-1)x+1\bigr)(nx+1)}=\frac1{(n-1)x+1}-\frac1{nx+1}$.

Answer 2

The series can be computed by using telescoping. We have $\frac{x}{[(n-1)x+1][nx+1]}=\frac1{(n-1)x+1}-\frac1{nx+1}$. Thus, the sequence of partial sums would be $s_n=\frac{nx}{nx+1}$. Hence, sum is equal to $$\begin{cases}0\ \text{when}\ x=0\\1\ \text{when}\ x\neq0\end{cases}$$. This easily shows non-uniform convergence, as the limit is discontinuous at $0$.