How can I calculate the value of the infinite sum $$\sum_{x=1}^\infty x^{-x}$$ By comparison to a geometric series, I know that it converges, but I don't know how to calculate an exact value. So far I have tried the following:
I have tried to turn it into a telescoping sum, but have had no luck
I tried adding a variable and differentiating it to create an easier sum
Any hints?