First, let me make it clear that I do not mean jokes of the
"abelian grape" variety. I take my cue from the following 
passage in A Mathematician's Miscellany by J.E. Littlewood
(Methuen 1953, p. 79):


  I remembered the Euler formula $\sum n^{-s}=\prod (1-p^{-s})^{-1}$;
  it was introduced to us at school, as a joke (rightly enough, and
  in excellent taste).


Without trying to define Littlewood's concept of joke, I venture
to guess that another in the same category is the formula


  $1+2+3+4+\cdots=-1/12$,


which Ramanujan sent to Hardy in 1913, admitting "If I tell you this 
you will at once point out to me the lunatic asylum as my goal."

Moving beyond zeta function jokes, I would suggest that the empty
set in ZF set theory is another joke in excellent taste. Not only
does ZF take the empty set seriously, it uses it to build the whole
universe of sets.

Is there an interesting concept lurking here -- a class of
mathematical ideas that look like jokes to the outsider, but which
turn out to be important? If so, let me know which ones appeal to
you.

Let "$\int$" denote $\int_0^x$. We want to find the solution to

$$\int f = f-1.$$

We simply "factor out" $f$, getting $1=\left(1-\int\right)f$. Thus, $f=(1-\int)^{-1}1$. 

Using the geometric series,

$$f=\left(1+\int+\iint+\iiint+\cdots\right)1=1+\int_0^x1~dx+\int_0^x\int_0^x1~dx+\cdots$$

Thus,

$$f=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots=e^x,$$

as expected. (This was told to me by Steve Miller)
