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Symbolic mathematical evaluations

Math is really fun, especially when is done symbolically.

# Overview This time we're taking a look at some interesting relations and identities for fractions, which will give us an insight of what is really going on, for example, in an infinite sum and how we can analyze it by evaluating it symbolically.
There is an useful and interesting identity for summing two fractions:
$$\frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd}$$

The question is: can we extend it to three fractions? What about ten? What about an infinite number of them?

Well, yes, this is possible, and it's actually quite easy to find a general formula to this. To give you a taste how this can be analyzed, let's sum four fractions (using a for the numerator and b for the denominator, just for illustration, but they can have different values):

$$\frac{a}{b} + \frac{a}{b} + \frac{a}{b} + \frac{a}{b} = \frac{b(b(b(a) + ab) + abb) + abbb}{bbbb}$$

Do you see the pattern? There is a beautiful recursive relation w…