I already asked a similar question on another post: What's the sum of $\sum \limits_{k=1}^{\infty}\frac{t^{k}}{k^{k}}$? There are no problems with establishing a convergence for this power series: $\sum_{k=1}^\infty \frac{2^{kx}}{e^{k^2}}$ but I have problems in determining its sum.
What's the sum of $\sum_{k=1}^\infty \frac{2^{kx}}{e^{k^2}}$?
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calculus
sequences-and-series
power-series
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3Substitute $t=2^x$, consider $\sum\limits_{k=1}^\infty \frac{t^k}{e^{k^2}}$ – 2012-03-02
2 Answers
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There is this Jacobi theta function: $ \vartheta_3 \biggl(\frac{i}{2} x \operatorname{ln} (2),\operatorname{e} ^{-1}\biggr) = \sum_{k = -\infty}^{\infty} \operatorname{e} ^{-k^{2}} 2^{k x} $ But you stopped half-way through, so yours is not such a common one. Yours is a "partial theta function"
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$\sum_{k=1}^{\infty}\frac{2^{kx}}{e^{k^{2}}} = -\frac{1}{2} + \frac{1}{2} \prod_{m=1}^{\infty} \left( 1 - \frac{1}{e^{2m}} \right) \left( 1+ \frac{ 2^x }{e^{2m-1} } \right) \left( 1 + \frac{1}{2^x e^{2m-1} }\right ). $
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0@Ragib: Thanks! – 2012-03-02