Is $0.1010010001000010000010000001 \ldots$ transcendental? Does anyone know if this number is algebraic or transcendental, and why?
$$\sum\limits_{n = 1}^\infty  {10}^{ - n(n + 1)/2}  = 0.1010010001000010000010000001 \ldots $$
 A: The number $0.1010010001000010000010000001\ldots$ is transcendental.
Consider following three Jacobi theta series defined by
$$\begin{align}
\theta_2(q) 
&= 2q^{1/4}\sum_{n\ge 0} q^{n(n+1)} = 2q^{1/4}\prod_{n=1}^\infty (1-q^{4n})(1 + q^{2n})\\
\theta_3(q)
&= \sum_{n\in\mathbb{Z}} q^{n^2} = \prod_{n=1}^\infty (1-q^{2n})(1+ q^{2n-1})^2\\
\theta_4(q)
&= \theta_3(-q) =
\sum_{n\in\mathbb{Z}} (-1)^n q^{n^2} = \prod_{n=1}^\infty (1-q^{2n})(1- q^{2n-1})^2\\
\end{align}
$$
and for any $m \in \mathbb{Z}_{+}$, $k \in \{ 2, 3, 4 \}$, use 
$\displaystyle D^m\theta_k(q)$ as a shorthand for 
$\displaystyle \left( q\frac{d}{dq} \right)^m \theta_k(q)$.
Based on Corollary 52 of a survey article Elliptic functions and Transcendence by M. Waldschmidt in 2006,

Let $i, j$ and $k \in \{ 2,3,4 \}$ with $i \ne j$. Let $q \in \mathbb{C}$
  satisfy $0 < |q| < 1$. Then each of the two fields
  $$
\mathbb{Q}( q, \theta_i(q), \theta_j(q), D\theta_k(q))
\quad\text{ and }\quad
\mathbb{Q}( q, \theta_k(q), D\theta_k(q), D^2\theta_k(q))
$$
  has transcendence degree $\ge 3$ over $\mathbb{Q}$

We know for any non-zero algebraic $q$ with $|q| < 1$, the three $\theta_k(q)$, in particular $\theta_2(q)$ is transcendental. Since 
$$\sum_{n=1}^\infty 10^{-n(n+1)/2} = \frac{\sqrt[8]{10}}{2} \theta_2\left(\frac{1}{\sqrt{10}}\right) - 1$$
and using the fact $\frac{1}{\sqrt{10}}$ and $\frac{\sqrt[8]{10}}{2}$ are both algebraic, we find the number at hand is transcendental.
