How to calculate the result of:
$$\frac{2012(1!)}{3!}+\frac{2012(2!)}{4!}+\frac{2012(3!)}{5!}+...+\frac{2012(2010!)}{2012!}$$
What is the theory used? Sequence of number? Please help me understand with clear step by step as I haven't learned about this. I only understand that there are some factorial in it.
$$\frac{2012(1!)}{3!}+\frac{2012(2!)}{4!}+\frac{2012(3!)}{5!}+...+\frac{2012(2010!)}{2012!}=$$$$2012\sum_{n=1}^{2010}\frac{n!}{(n+2)!}=2012\sum_{n=1}^{2010}\frac{1}{(n+1)(n+2)}$$Because this is a telescoping series, we can easily solve now.