prove this equality

Question

First, I try to use the Taylor series but it doesn't work well. And someone said I can use fundamental theorem of calculus but I don't see it.

can anyone give me a good hit?

thanks

Answer 1

Here is a start:

Let $f_n( x) =D_n(x^{n-1}e^{1/x}) $, where $D_n$ is the $n^{th}$ derivative.

By Leibnitz's formula $D_n(a(x)b(x)) =\sum_{k=0}^n \binom{n}{ k} D_k(a(x))D_{n-k}(b(x)) $. (This is readily proved by induction using the regular product rule for derivatives.)

Then get the appropriate derivatives.

Note that many of the derivatives may be zero, which might simplify things.