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.)

Mathematicais able to do the Aitken "trick", but the function does not seem to be explicitly advertised. Try`SequenceLimit[(*sequence*), Method -> {"WynnEpsilon", "Degree" -> 1}]`

– J. M. May 15 '11 at 14:12