Summation of Hardy's series

Question

I recently found this series from an Hardy work:

$$\sum_{n=0}^{+\infty}(-1)^nx^{2^n}=x-x^2+x^4-x^8+\dots$$

For what values of $x$ does it converge ? Can we use some summation technique to sum it where it should diverge ? For $x=1$ this looks like Grandi's series so I would say it's values should be $\frac 12$ or am I wrong ?

Answer 1

Partial answer.

For $F_R(x)=Rx+R^2x^2+R^3x^4+R^4x^8+\cdots$, $R\neq1$, we have $F_R(x)-RF_R(x^2)=Rx$. Thus $F_R(1)$ belongs to the elementary Ramanujan class $R$ and is summable to $R/(1-R)$ (definition, also here). For $R=-1$, the elementary Ramanujan sum of the diluted Grandi series $1-1+0+1+0+0+0-1+\cdots$ is actually $1/2$.