Calculation of a series using series definition I have to calculate the series $$\sum_{k=2}^\infty \left(\frac 1k-\frac 1{k+2}\right)$$
Using the definition: $$L = \lim_{n\to\infty}S_n=\lim_{n\to\infty}\sum_{k=0}^na_k$$ 
Obviously $\lim_{n\to\infty} (\frac 1n-\frac 1{n+2})=0$, but I don't think that this is the right way to calculate the value of the series.
 A: Let us use $1/k=\int_{0}^1 t^{k-1} dt$. Then $$S=\sum_{k=2}^{\infty} \left (\frac{1}{k}-\frac{1}{k+2} \right)=
\int_{0}^{1} \sum_{k=2}^{\infty} ~[t^{k-1}- t^{k+2}]~dt =\int_{0}^{1} \frac{t-t^3}{1-t} dt=\int_{0}^{1} (t+t^2) dt=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}.$$
A: As suggested from the comment by JMoravitz, you should write down the series explicitly and see if you can see something interesting:
$$ \sum_{2}^\infty \big( \frac{1}{k} - \frac{1}{k+2} \big) = \bigg(\frac{1}{2} - \frac{1}{4} \bigg) + \bigg(\frac{1}{3} - \frac{1}{5} \bigg) + \bigg(\frac{1}{4} - \frac{1}{6} \bigg) + \bigg(\frac{1}{5} - \frac{1}{7} \bigg) + \bigg(\frac{1}{6} - \frac{1}{8} \bigg) + \cdots + \bigg(\frac{1}{n} - \frac{1}{n+2} \bigg) + \bigg(\frac{1}{n+1} - \frac{1}{n+3} \bigg) + \bigg(\frac{1}{n+2} - \frac{1}{n+4} \bigg) + \bigg(\frac{1}{n+3} - \frac{1}{n+5} \bigg) + \cdots $$
From here, you can see which terms are cancelling with one another and which one is left.... Essentially you should see that all the terms will cancel with each other except the $\frac{1}{2}$ and $\frac{1}{3}$. Therefore, the answer is $\frac{5}{6}$. 
$\textbf{Another, Different way}$ that uses limit like you wanted to is to find the partial sum! You first compute, $S_2 = \frac{1}{2} - \frac{1}{4} $, $S_3 = S_2 + \bigg( \frac{1}{3} - \frac{1}{5} \bigg)$ and so on... What you will find (and you can prove this fact by induction) is that 
$$ S_n = \dfrac{5n^2 + 3n -8}{6(n+1)(n+2)}$$
Now, $$ \lim_{n \to \infty} \dfrac{5n^2 + 3n -8}{6n^2 + 18n + 12} = \frac{5}{6} $$
A: Hint:
Try writing out the first few partial sums and look to see if anything can cancel.
Even more explicitly, looking ahead to somewhere in the middle, you will have if you expand out the summation the following:
$$\dots +\left(\dfrac{1}{50}-\color{blue}{\dfrac{1}{52}}\right)+\left(\dfrac{1}{51}-\color{red}{\dfrac{1}{53}}\right)+\left(\color{blue}{\dfrac{1}{52}}-\dfrac{1}{54}\right)+\left(\color{red}{\dfrac{1}{53}}-\dfrac{1}{55}\right)+\dots$$
Now, notice the colors I used and think about why I put colors there.  Where else in the series does something like this happen?  What does this imply in the end will be left over if we take a partial sum?  What does this imply in the end will be left over if we consider the limit of partial sums?
The more general name for this property is "Telescoping."  We refer to this series as a "Telescoping Series."
