Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?

Is there any closed form expression for the following serie?

$$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$

Or at least a proof that it is an irrational number. The context of this problem is given by the following link:

https://math.stackexchange.com/questions/2270730/whats-the-limit-of-sqrt2-sqrt2-sqrt2-sqrt2-sqrt2-sqrt2

In which it is proposed the problem of finding a closed form for the following nested radical:

$$R = \sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + …}}}}}} $$

whose signs follow the pattern: $10100100010000…$, where $1 = +$, $0 = -$. The expression for that radical is given by:

$$2\cos\left[\frac{\pi}{6}\left(1 – 2\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}\right)\right]$$

And thus the question of obtaining a closed form expression for that series arises. As noted in the post, it resembles a Jacobi theta’s function, except that the range for $k$ to sum over doesn’t match.

Answer

Attribution
Source : Link , Question Author : John Finkelstein , Answer Author : Community

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