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if $$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$

then is there a way to deduce this sum:$$\sum_{n=0}^\infty z^{n^3}$$ from the

above Identitie or any visual proof if it was existed ?

Note: $z$ is a complex variable

Thank you for any help

Zeraoulia rafik
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    I'd sort of assume the answer is "no" - I doubt there is a closed form. The theta function can handle the case of summing $z^{n^2}$ - but it is itself a special function, and is little better than just treating the series abstractly, and evaluating numerically. The closed form for summing $z^n$ can be found with generating functions, but that approach will only yield answers for algebraic functions. I can't prove the latter sum isn't algebraic, but it seems very unlikely. – Milo Brandt Jun 22 '15 at 01:12
  • many mathematicians used the above geometric serie, but what about the second at a least a visual proof .? – Zeraoulia rafik Jun 22 '15 at 01:15
  • but theta function can't handle the case of z^n^3 – Zeraoulia rafik Jun 22 '15 at 01:34
  • See my answer to this question. – Lucian Jun 22 '15 at 03:40

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