A few months ago, I stumbled a pond a new problem solving idea. It is called the telescope method. I did know about it but I just used it subconsciously. I want to present a rather easy problem that can be solved with this method. In 1672 Huygens asked Leibnitz to solve this problem, which he did using the same method.
Now there is a nice trick that you can use:
This has a great benefit for infinite sums. We add this into the equation above and get:
Lets se what happens to the first few n. 1/1-1/2+1/2-1/3+1/3-14+1/4-1/5. As you can see, everything except the 1/1 cancels itself out. 1/1+(-1/2+1/2)+(-1/3+1/3)+(-14+1/4)+(-1/5… =1/1+0+0+0+0….
We have 1=1/2x. Simple algebra gives us x=2, which is the solution to the problem.