TITLE: Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?
QUESTION [99 upvotes]: Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$?
(I think it is.)  If so, how would one prove this?  (To confirm:  This is the power
series for $e^x$, except with the denominator replaced by $\sqrt{n!}$.)

REPLY [4 votes]: Although the following does not provide another proof (perhaps it is possible to attempt one on this basis) I found it nice to see the following pictures.
Let's take from the series $f(x) = \sum_{k=0}^\infty {x^k \over \sqrt{k!}}$ the following variants in the same spirit as we have the hyperbolic and trigonometric series from the exponential-series:
$$\begin{array}{}
 \small \exp_{\tiny \sqrt{\,}}(x) &=& f(x) \\
 \small \cosh_{\tiny \sqrt{\,}}(x) &=& \sum_{k=0}^\infty {x^{2k} \over \sqrt{(2k)!}} \\
 \small \sinh_{\tiny \sqrt{\,}}(x) &=& \sum_{k=0}^\infty {x^{2k+1} \over \sqrt{(2k+1)!}} \\
\small  \tanh_{\tiny \sqrt{\,}}(x) &=& { \sinh_{\tiny \sqrt{\,}}(x)\over \cosh_{\tiny \sqrt{\,}}(x)  } \\
\small  \cos_{\tiny \sqrt{\,}}(x) &=& \sum_{k=0}^\infty (-1)^k {x^{2k} \over \sqrt{(2k)!}} \\
\small  \sin_{\tiny \sqrt{\,}}(x) &=& \sum_{k=0}^\infty (-1)^k {x^{2k+1} \over \sqrt{(2k+1)!}} \\
\end{array}$$ 
The answer to your question is equivalent to say, that always (="for real $x$")          

$\small \cosh_{\tiny \sqrt{\,}}(x)$ is larger than $\small \sinh_{\tiny \sqrt{\,}}(x) $ $\qquad \qquad$ or that
$\small \mid \tanh_{\tiny \sqrt{\,}}(x) \mid \lt 1$


To illustrate this I've plotted the $\sinh_{\tiny \sqrt{\,}}$ and $\cosh_{\tiny \sqrt{\,}}$-curves:              
 
This gives surely an extremely familiar impression...            
The $\tanh_{\tiny \sqrt{\,}}$-curve looks completely familiar too:
 
and the image suggests, that indeed the absolute value of $\small  \tanh_{\tiny \sqrt{\,}}(x) $ very likely is smaller than $1$ for all real $x$.

However, things are different for the $\sin_{\tiny \sqrt{\,}}$ and $\cos_{\tiny \sqrt{\,}}$ curves - they deviate strongly from the nicely periodic common trigonometric functions:                  
 
and combined they do not give a circle, but some ugly thing, strongly distorted (y-axis by $\small \cos_{\tiny \sqrt{ \,} }(\phi)$, x-axis by $\small \sin_{\tiny \sqrt{ \,} }(\phi)$, $\phi$ from $-5$ to $+5$) :