Mastering Partial Fractions for Solving Advanced Summation Problems

[embassyhill]

Homework Statement

Homework Equations

The Attempt at a Solution

Obviously I don't need a solution because it's right there. What I need to understand is what happened after the third equation sign and more importantly, how would I learn to solve these kinds of problems on my own. I looked at Wikipedia and even though it was helpful, I still don't understand this specific exercise for example. I would be grateful if someone could point me at a resource on this topic.

 

[gb7nash]

So you're wondering how to go from:

$$\frac{1}{2}(\frac{1}{1}-\frac{1}{3}) + \frac{1}{2}(\frac{1}{3}-\frac{3}{5}) + ... + \frac{1}{2}(\frac{1}{2n-1}-\frac{1}{2n+1})$$

to:

$$\frac{1}{2}(1-\frac{1}{2n+1})$$

Factor a 1/2 out of all the terms and then add the stuff inside. Something interesting happens. For example:

(1/2)(a+b) + (1/2)(c+d) = (1/2)(a+b+c+d)

 

[embassyhill]

Ok so everything but 1/1 and -1/(2n+1) cancel each other out. Now I get this exercise but what about the bigger question? Are there any tricks to learn here (about the sigma) or is it just about using your knowledge of algebra?

 

[gb7nash]

It depends on the problem, but a lot of times you can use mathematical induction to prove summation results. There's an example of it on wiki:

http://en.wikipedia.org/wiki/Mathematical_induction

This assumes that you know what you're trying to prove ahead of time (like in the case of your example). However, if you want to know a general way of finding the sum of n terms, there's no standard way, as far as I know. The best you can usually do is some trick like the one I showed you, or see if you're looking at a geometric sum. Otherwise, memorize these:

http://en.wikipedia.org/wiki/Summation

 

[awkward]

Partial fractions:

$$\frac{1}{(2x-1)(2x+1)} = \frac{1/2}{2x-1} - \frac{1/2}{2x+1}$$