## You can count on power series

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Bonus track session in CCMB 217, Wed Apr 25, 2001, 7-9pm

You have seen that power series are useful for all kinds of calculus stuff:
integration, limits, differential equations. But you may be surprised that
they are very helpful even when it comes to something as basic as counting!
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 C_{n}H_{2n+1}OH ?
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.

Last modified: Apr 23, 2001