At the beginning of the 19th century a young mathematician named Niels Abel wrote that

"divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever". 

In this talk, without any shame, we will see what divergent series are, how to deal with them and how to use them to extract sensible results in a variety of different problems, from the Stirling approximation of n! to tunnelling problems in quantum mechanics and, to add a little colour, even to the study of 

rainbows.