TITLE: Useful tricks in experimental mathematics
QUESTION [30 upvotes]: There are a few computational tricks which are useful in experimental mathematics.
These tricks are mostly very elementary and often only given as exercices in books.
A typical example is the following:
Suppose that a sequence $s_0,s_1,s_2,\dots$ converges exponentially fast. Then the sequence $t_i=s_i-\frac{(s_{i+1}-s_i)^2}{s_{i+2}-2s_{i+1}+s_{i}}$
converges (generally) faster and has the same limit. Having only access to a few initial
terms of a sequence which seems to converge quickly, this trick improves thus guesses concerning the limit.
This suggests two questions:

Is there a nice book/article containing a list of useful tricks "ready for use"?
What tricks are useful for you?

For clarity let me state that I do not count Euclid's algorithm, LLL or such things as 
tricks. they are already implemented and ready for use in computer-algebra systems. (A nice
book concerning tricks might have however also ulterior chapters mentioning such useful algorithms and describing them very briefly.)

REPLY [2 votes]: For the first question ("is there a nice book/article..."), I think the answer ie Yes: Sanjoy Mahajan's Street-Fighting Mathematics, which also exists in a free CC version, summarizes a good number of useful tricks and meta-tricks, some well-known, some less so.