Complex value of a divergent series

Question

Given the series: $$S=\sum_{k=1}^{\infty}\frac{2^k}{k^2}$$ the sum obviously doesn't converge. 'Maple' gives for the value of the series: $$S(a)=\sum_{k=1}^{\infty}\frac{a^k}{k^a}$$ $S(a)=Li_a(a)$ with $Li$ polylogarithmic function. For $a=2$, for example, $Li_2(2)\approx 2.46-2.17i$. The question is: if the series is not convergent and is a sum of real number, how is it possible to get a complex value for $S(a)$? Thanks.

Answer 1

Lets take a look at $S_i(x) = \sum\limits_k \frac 1 {k^i} x^k$. This is a power series with radius of convergence $\lim\limits_{k \to \infty}\frac { (\frac 1 {k+1})^i} {(\frac 1 {k})^i} = 1$. So the representation $S_i(x) = \operatorname{Li}_i(x)$ is only valid for $|x| \leq 1$, and not for $x=2$.

However, through analytic continuation, we can extend this function to $|x| > 1$, but are not guaranteed a real valued output for real valued input.

Answer 2

Maple can make mistakes. I know of a similar error which occurs when computing the Laplace transform of $e^{t^2}$ in Wolfram Alpha. They claim the existence of $$\int_0^\infty e^{t^2} e^{-s t} dt $$ which clearly diverges for all real $s$. See here.