For instance, how many possibilities are there to pay an amount of $1.77 ? Available are coins at 1, 5, 10, 25 cents, and $1 coins and bills. But we have to take into account that there are many different quarters (for lack of knowledge how many state quarters have already been issued, I'll assume we have 15 different quarters available), and there are 3 types of $1.- currency: old and new coins, and George Washington.
You will understand, in leisurely discussion how this problem can be encoded in power series.
But, as you may already know from comparing the series method with l'Hopital's method of calculating limits, there is one remarkable feature about power series, namely: The tougher the problem, the happier the power series. So here's a sophisticated problem that makes the power series really powerful and happy:
How many structurally different compounds (so-called isomers) could there be that correspond to the chemical sum formuls CnH2n+1OH ? The answer will be given in terms of a power series problem that you can solve. Unlike the coin example, we will have to skip how to translate the original problem into power series (that would involve another, more advanced, mathematical concept). But it is still a feast to see how power series produce the answer, seemingly by miracle. -- And just in case you have forgotten or never learned the chemistry required to understand the very question here, don't worry: this will be the very easy part explained in the beginning.