I was reading a blog post earlier about the Sophomore's Dream and a question came to mind:
Say we wanted to find a definite integral that gives the following result
$$\sum_{n=1}^\infty \left(\frac{a}{n}\right)^n=a+\left(\frac{a}{2}\right)^2+\left(\frac{a}{3}\right)^3+\cdots$$
My guess would be that we'd try to find a function $f(a)$ such that
$$\int^\beta_\alpha f(a)\,\mathrm{d}a = \sum_{n=1}^\infty \left(\frac{a}{n}\right)^n$$
assuming that it's indeed $f(a)$ we want (and not of some other variable)...
Is this an integral equation (I'm quite unfamiliar with them), and if so, is there a popular method to solve those of this form?
Or could we just use the FTC or something to solve for $f$?
Otherwise, how would you go about deriving such an indefinite integral?