Continuity of $f(x) = \frac 1 x - \frac 1 {x+1} + \frac{1}{x+2} - \cdots$ 
Is function 
  $$f(x) = \frac 1 x  - \frac 1 {x+1} + \frac{1}{x+2} - \frac 1{x+3} + \cdots$$
  continuous on $(0, \infty)$ ?

I think I should extend the continuity of each term to $f$ but I cannot show that series converges uniformly.

$$\left( \frac 1 x - \frac 1 {x+1} \right) + \left( \frac 1 {x+2} - \frac 1 {x+3} \right) + \cdots + \left( \frac 1 {x+2n} -\frac 1 {x+2n-1} \right)$$ $$=\frac 1 {x(x+1)}+\frac 1 {(x+2)(x+3)}+\cdots +\frac 1 {(x+2n) (x+2n+1)}$$ $$\leq \frac 1 {x(x+1)}+\frac 1 {2^{2}}+ \frac 1 {4^{2}}+\cdots+\frac 1 {(2n)^{2}} $$. 
This gives uniform convergence of the partial sums. 
