Fibonacci series $\lim_{x\to +\infty}\sum_{n=1}^{\infty}\frac{a_n}{n!}x^n$ Let ${a_n}$ be a Fibonacci series $a_0=a_1=1$ and $a_{n+2}=a_n+a_{n+1}$ for every $n \geq 0$.
How is calculated the following limit  $$\lim_{x\to +\infty}\sum_{n=1}^{\infty}\frac{a_n}{n!}x^n\;,\;x\in\mathbb{R}$$
Any hint would be appreciated.
 A: Since the $a_n$ are all at least 1, for $x \geq 0$ we have
$$
  \sum_{n\geq 1} \frac{a_n}{n!} x^n
  \geq \sum_{n \geq 1} \frac{x^n}{n!}
  = \exp(x) - 1.
$$
A: The exponential generating function for the Fibonacci numbers is given by
\begin{align}
S(t) = \sum_{n=0}^{\infty} F_{n} \, \frac{t^{n}}{n!} = \frac{e^{\alpha t} - e^{\beta t}}{\alpha - \beta}
\end{align}
where $2 \alpha = 1 + \sqrt{5}$ and $2 \beta = 1-\sqrt{5}$. As $t \rightarrow \infty$ it is easy to seen that $S(t) \rightarrow \infty$.  
A: the $n^{th}$ term of a fibonacci sequence is $A\alpha^n + B\beta^n$, so
$$
 \sum_{n=0}^{\infty} F_{n} \, \frac{x^{n}}{n!} =  \sum_{n=0}^{\infty} (A\alpha^n + B\beta^n) \, \frac{x^{n}}{n!} = A e^{\alpha x} + B e^{\beta x}
$$
A: Although you can determine that the limit is infinite without actually evaluating the series, it’s not too hard to do the latter as well. We might as well set $a_0=0$; the desired function is then
$$g(x)=\sum_{n\ge 0}\frac{a_nx^n}{n!}\;.$$
Differentiating twice, we find that
$$g'(x)=\sum_{n\ge 0}\frac{a_{n+1}x^n}{n!}$$
and
$$g''(x)=\sum_{n\ge 0}\frac{a_{n+2}x^n}{n!}\;.$$
But then
$$\begin{align*}
g''(x)&=\sum_{n\ge 0}\frac{a_{n+2}x^n}{n!}\\
&=\sum_{n\ge 0}\frac{(a_n+a_{n+1})x^n}{n!}\\
&=g(x)=\sum_{n\ge 0}\frac{a_nx^n}{n!}+g(x)=\sum_{n\ge 0}\frac{a_{n+1}x^n}{n!}\\
&=g(x)+g'(x)\;,
\end{align*}$$
a differential equation that can be solved by standard methods. If $\varphi=\frac12(1+\sqrt5)$ and $\widehat\varphi=\frac12(1-\sqrt5)$, then
$$g(x)=\frac1{\sqrt5}\left(e^{\varphi x}-e^{\widehat\varphi x}\right)\;.\tag{1}$$
Now $\varphi>1$ and $|\widehat\varphi|<1$, so for large $x$ $(1)$ is dominated by the first term and blows up as $x\to\infty$.
The function $g$ is the exponential generating function of the Fibonacci sequence.
