Question: How to find the formula for the sum of fractions like this?
$$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\ldots+\frac{1}{n\times (n+1)}=$$

Answer: 
You can write $\dfrac 1{n(n+1)} = \dfrac 1n - \dfrac 1{n+1}$. You get a lot of cancellation after which the expression equals $1 - \dfrac{1}{n+1}$.
Since $$\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$$ for all $k$, you sum telescopes to 
$$1 - \frac{1}{n+1}.$$