I’m a big fan of non-rigorous arguments, especially in calculus and analysis. I think there should be a book cataloging all the beautiful, morally-true-but-not-actually-true proofs that mathematicians have advanced, but until that time I’ll try to at least catalog a few of them on my blog.
This first one is Euler’s original argument for the equality of two expressions (both of which happen to define ):
I’ll also sketch how this can be made rigorous in non-standard analysis.
The argument is as follows: The limit is equal to , where is infinitely large. By the binomial theorem, this is:
Since is , this is the sum as ranges from to of:
Now, if is infinitely large, this term is so small that it may be neglected. On the other hand, if is finite, then for . Therefore
and the whole sum is equal to
Now, I’ll sketch how to make this rigorous in non-standard analysis. This is from Higher Trigonometry, Hyperreal Numbers, and Euler’s Analysis of Infinities by Mark McKinzie and Curtis Tuckey, which is the best introductory article on non-standard analysis that I’ve read.
In non-standard analysis, one extends the real numbers to a larger field which contains all the reals, but also a positive which is less than every positive real (and hence also a number which is greater than every real). For every function , there is a function , and the ‘s satisfy all the same identities and inequalities formed out of composition that the ‘s do. (For example, for all hyperreal .) For that reason, I’ll often omit the . The range of is called , the hyperintegers. Since , the same is true in the hyperreals and there are therefore infinite hyperintegers.
We call a nonzero hyperreal infinitesimal if is less than every positive real. We say that and are close (written ) if is infinitesimal. We say that is infinite if is infinitesimal (equivalently, if is greater than every real). We say that is finite if it’s not infinite (equivalently, if is less than some real).
Let be a sequence of hyperreals. We say that it is determinate if whenever and are infinite
The Summation Theorem can then be proven: If and are two determinate sequences such that for all finite , then for all infinite .
By appropriately replacing “equals” with “is close to”, Euler’s argument above may now be adapted to prove that for all infinite and ,
(the sequence may be proved determinate by comparison with the geometric sequence, which is easily shown determinate). By a transfer principle, this may in turn be used to prove that (in the regular reals).