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What are the results of this: $$\sum_{i = 1}^{n}a^{i^{2}}$$ and this: $$\sum_{i = 1}^{n}a^{i^{3}}$$ expressed in terms of $n$, with $a$ being a predefined constant? Is there any general rule to calculating the sums of series like these for an arbitrary power of $i$?

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No, there is no formula for these sums appreciably simpler than just writing them as sums. –  Gerry Myerson Jan 16 '13 at 10:38

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For the first infinite series, what you want to learn about is Jacobi Theta Functions.

See, for example, the result in Wolfram|Alpha.

I don't know of anything corresponding to the second series.

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