Showing that the infinite series $\sum_1^\infty \left(\frac {1}{n} - \frac{1}{n+2}\right)$ is convergent

Question

If we consider an infinite series witht the $n^{th}$ term $$a_n= \frac {1}{n} - \frac{1}{n+2}$$ for $n\ge1$

I am used to calculate the value to which a geometric series converges by looking at the coefficient. However, here I do not have the coefficient but the series itself. How could I show that this infinite series converges and calculate its sum?

Answer 1

That's not a geometric series, but it converges. Evaluate the partial sum:

$$\require{cancel}\sum_{k=1}^na_k=\left(1-\cancel{\frac13}\right)+\left(\frac12-\cancel{\frac14}\right)+\left(\cancel{\frac13}-\cancel{\frac15}\right)+\ldots+\left(\cancel{\frac1n}-\frac1{n+2}\right)=$$

$$=\frac32+\frac1{n+2}\xrightarrow[n\to\infty]{}\frac32$$