continuity of the function $f(x)=\lim_{n\to \infty}\sum_{k=0}^{n-1} \dfrac{x}{(kx+1)[(k+1)x+1]}$

Question

I need to check the continuity and differentiability of the function $f(x)$ at $x=0$ where, $$f(x)=\lim_{n\to \infty}\sum_{k=0}^{n-1} \dfrac{x}{(kx+1)[(k+1)x+1]}.$$

I tried to check the domain of convergence convergence, and applied the ratio test, but the ratio is coming one and hence unable to conclude anything. If the function converges uniformly the also we can say the limit will be continuous but I am unable to check the convergence. How will I check the continuity and differentiability of this function?

Answer 1

Hint:

Write the numerator as

$$(k+1)x+1-(kx+1)=?$$ to form a telescoping series