Suppose that $f(x):=\sum_{k\geq 0}(-1)^ke^{-(2k+1)^2x}$ has a holomorphic continuation to a neighborhood of $0$, that is, $f(x)=\sum_{n\geq 0}a_n x^n$ for $x> 0$ small. I want to know the value of $a_1$. Let $z=e^{-x}$, $g(z):=\sum_{k\geq 0}(-1)^k z^{(2k+1)^2}$; then, formally, $g'(1)$ is what we want to compute. Note that $1$ is on the circle of convergence of $g$ so differentiating term-wise is not guaranteed.