# Closed-form expression for infinite series related to a Gaussian

Consider the following infinite series, where $$x$$ is indeterminate and $$r$$ is held constant:

$$\displaystyle 1 + \frac{x}{r} + \frac{x^2}{r^2} + \frac{x^3}{r^3} + ...$$

It is relatively easy to see that the above, for $$\frac{x}{r} < 1$$, converges to

$$\displaystyle \frac{1}{1-\frac{x}{r}}$$

Now suppose we modify the above to this:

$$\displaystyle 1 + \frac{x}{r} + \frac{x^2}{r^{2^2}} + \frac{x^3}{r^{3^2}} + ...$$

which we can rewrite as

$$\displaystyle 1 + \frac{x}{r} + \frac{x^2}{r^4} + \frac{x^3}{r^9} + ...$$

Does there then exist a well-known, closed-form expression for this series?

If not for general $$r$$, then perhaps for certain special values of $$r$$? For example, if we set $$r=e$$ above, then we get

$$\displaystyle 1 + \frac{x}{e} + \frac{x^2}{e^4} + \frac{x^3}{e^9} + ...$$

which we can rewrite as

$$\displaystyle 1 + xe^{-1^2} + x^2e^{-2^2} + x^3e^{-3^2} + ...$$

So that we can see that for x=1, this becomes a series of evenly spaced points on a Gaussian function.

Does there exist a closed-form expression for any of these?

Interestingly, if we change the definition slightly, we get something related to the Jacobi theta function.

$$\displaystyle 1 + \frac{x}{r} + \frac{x^2}{r^4} + \frac{x^3}{r^9} + ...$$

We can make the following substitutions:

$$q = \frac{1}{r}$$

$$x = \exp(2\pi i z)$$

to obtain

$$\displaystyle 1 + q\exp(2\pi i z) + q^4\exp(4\pi i z) + q^9\exp(6\pi i z) + ...$$

$$= \sum_0^\infty q^{n^2} \exp(2\pi i n z)$$

If we simply change the bottom bound from $$0$$ to $$\infty$$, we get

$$\theta_3(z;q) = \sum_{-\infty}^\infty q^{n^2} \exp(2\pi i n z)$$

So it is easy to write the Jacobi theta function in terms of the function I described; it is probably possible to write it the other way as well.

Your original series is $$f(x, r) =\sum_{n=0}^{\infty} \dfrac{x^n}{r^{(n^n)}}$$.

This is not the same as $$g(x, r) =\sum_{n=0}^{\infty} \dfrac{x^n}{(r^n)^n} =\sum_{n=0}^{\infty} \dfrac{x^n}{r^{n^2}}$$.

Also, you went from $$\displaystyle 1 + \frac{x}{e} + \frac{x^2}{e^4} + \frac{x^3}{e^9} + ...$$ to $$\displaystyle 1 + x \cdot e^{-1^2} + x \cdot e^{-2^2} + x \cdot e^{-3^2} + ...$$, somehow losing the exponent of $$x$$.

• Thanks, edited the typos – Mike Battaglia Oct 15 '18 at 18:18