I ask that anyone who has had multivariable calculus please stop right now and try to work out this problem.
$$\int_0^1\int_0^1\frac{dx\,dy}{1-xy}.$$
Hint: The integrand looks familiar!
Solution.
Notice that the integrand is $\frac{1}{1-xy}$. This should suggest a geometric series, as it looks to be of the form $\frac{1}{1-r}=\sum_{n=0}^\infty r^n$, since $|x|,|y|<1$. So, let's look at it like this.
$$\int_0^1\int_0^1\frac{dx\,dy}{1-xy}=\int_0^1\int_0^1\sum_{n=0}^\infty \left(xy\right)^n dx\,dy=\sum_{n=0}^\infty\int_0^1\int_0^1x^ny^ndx\,dy=$$
$$\sum_{n=0}^\infty\left(\int_0^1x^ndx\right)\left(\int_0^1y^ndy\right)=\sum_{n=0}^\infty\left(\frac{1}{n+1}\right)^2=\sum_{n=1}^\infty\frac{1}{n^2}.$$
Now, if only we knew what this last sum came out to be, we'd be able to finally answer the question definitively. However, the typical calculus techniques do not work to evaluate this sum, and so we will need to work a little harder. This sum will be the subject of a future post.