The question I've been posed is:
Show that the following series converges, and compute its value
$$\sum_{k=1}^\infty \frac{1}{k(k+2)}$$
From this I decided to use partial fractions to put into the form:
$$\frac{1}{2}\cdot\left(\frac{1}{k}-\frac{1}{k+2}\right)$$
And from this I noticed that this is in the form of a telescoping series which I think would cancel down to:
$$\frac12\cdot\left(1+\frac12\right)= \frac{3}{4}$$
So I've got to this point, but I don't think what I've worked out is substantial enough to prove what I've been asked.
Would anyone mind giving any tips to make my working more thorough.
Best Answer
Note that$$\frac1k-\frac1{k+2}=\left(\frac1k-\frac1{k+1}\right)+\left(\frac1{k+1}-\frac1{k+2}\right).$$This will give you two telescoping series. Can you take it form here?