Does the series $\sum_{n\ge1}\int_0^1\left(1-(1-t^n)^{1/n}\right)\,dt$ converge? Here is a question that I've been working on, a few years ago. I do know how to solve it but I am convinced that it deserves at least another point of view ...
I will post my own solution soon (within a week, at most) and I hope that - meanwhile - other people will suggest various approaches.
Consider, for all $n\in\mathbb{N^\star}$ :
$$u_n=\int_0^1\left(1-(1-t^n)^{1/n}\right)\,dt$$
Does the series $\sum_{n\ge 1}u_n$ converge ?

Using Euler's Beta function, we are simply asked about the convergence of
$$ \sum_{n\geq 1}\left(1-\frac{\Gamma\left(1+\tfrac{1}{n}\right)^2}{\Gamma\left(1+\tfrac{2}{n}\right)}\right) $$
where $\Gamma(1+x)$ is a regular function in a neighbourhood of $x=0$. Since
$$ \log\Gamma(1+x) = -\gamma x+\frac{\zeta(2)}{2!}x^2-\frac{\zeta(3)}{3}x^3+\frac{\zeta(4)}{4}x^4-\ldots $$
we have that 
$$ 1-\frac{\Gamma\left(1+\tfrac{1}{n}\right)^2}{\Gamma\left(1+\tfrac{2}{n}\right)} \sim \frac{\pi^2}{6n^2} $$
so the given series is convergent for sure. Geometrically, we are saying that if $A_n$ is the area of the region in the $[0,1]\times[0,1]$ square given by $x^{n+1}+y^{n+1}\leq 1 \leq x^n+y^n$, then
$$ A_1+2A_2+3 A_3+\ldots $$
is a convergent series. That can be proved by elementary means, too.
