Mathematical tricks for Real Analysis

I am currently studying computer science (first year) and one of our first courses is Real Analysis. What I'm struggling with the most are all those kinds of "tricks" that you supposedly should be aware of. I'll give an example: $$\sum_{n=1}^\infty \ \frac{1}{4n^2-1}$$ Finding out that this series converges with the comparison test is fairly understandable, but we had to find the sum too. How am I supposed to know that I could get a telescoping sum with partial fraction decomposition?
I know doing a lot of exercises will sharpen my intuition, but the problem is that it already takes so much time to do just two or three of these exercises (a couple of hours). It seems to me like I am missing basic mathematical tools (like partial fraction decomposition) or a better understanding of algebra. Most of my collegues struggle with similar problems. Do you know any good resources, books or exercises that specifically train all those "tools"?

• – JMoravitz Nov 17 '16 at 3:47
• thanks, I already saw that post :) There are some nice suggestions in there but not really any material to further exercise and develope any skill in "using" these tricks. – Alessio Eberl Nov 17 '16 at 3:53
• I don't want to discourage you; however, I would say there is no trick in pure mathematics. By "trick" I mean the shallow tools that get people over a math exam. Trick-minded thinking works only for technique-based math, I am afraid. Though I am not sure your Real Analysis course is at what level, a reasonable guess is that at least one should be able to understand an epsilon-delta proof... – Megadeth Nov 17 '16 at 3:53
• Good question. Somehow I immediately thought of telescoping sums, but not because I knew it would work, just because I know it is a solvable sum for which there must be a simple trick, and telescoping sums is one of the few tricks for sums (the other one I know is the geometric sum formula). [It also looked like a "contrived" problem that was factorable.] I think the main important math trick is being able to multiply and divide by the same thing, and also add and subtract the same thing, i.e., $a = a+b-b$. That alone can take you almost everywhere in a real analysis class. – Michael Nov 17 '16 at 3:56
• Yeah that's exactly the problem.. I get the proofs. I'm usually a pretty logical thinker, but things like the "$+1-1$" "tricks" leave me in speechless. It's so easy yet in most cases I wouldn't be able to come up with it myself. – Alessio Eberl Nov 17 '16 at 3:58