I've been trying to find an asymptotic expansion of the following series

$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$

and

$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n(n!{\sqrt{n}}) }$$

around $+\infty$, in the from

$$\exp(x^2)\Big(1+\frac{a_1}{x}+\frac{a_2}{x^2} + .. +\frac{a_k}{x^k}\Big) + O\Big(\frac{\exp(x^2)}{x^{k+1}}\Big)$$

where $x$ is a positive real number. As far as I progressed, I obtained only

$$C(x) = \exp(x^2) + \frac{\exp(x^2)}{x} + O\Big(\frac{\exp(x^2)}{x}\Big).$$

I tried to use ideas from http://math.stackexchange.com/questions/484367/upper-bound-for-an-infinite-series-with-a-square-root?rq=1, http://math.stackexchange.com/questions/115410/whats-the-sum-of-sum-limits-k-1-infty-fractkkk, http://math.stackexchange.com/questions/378024/infinite-series-involving-sqrtn?noredirect=1&lq=1, but I was unable to make them work in my case.

Any suggestions would be greatly appreciated!

(If someone has a solid culture in this kind of things, is there are any specific names for $C(x) $ and $L(x) $ ?).

PS:

This question was asked on the math.SE but was closed as duplicate of http://math.stackexchange.com/questions/2117742/lim-x-rightarrow-infty-sqrtxe-x-left-sum-k%ef%bc%9d1-infty-fracxk/2123100#2123100. However, the latter question provides only the first term of the asymptotic expansion and does not address sufficiently the problem considered here.

I've been trying to find an asymptotic expansion of the following series

$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$

and

$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n(n!{\sqrt{n}}) }$$

around $+\infty$, in the from

$$\exp(x^2)\Big(1+\frac{a_1}{x}+\frac{a_2}{x^2} + .. +\frac{a_k}{x^k}\Big) + O\Big(\frac{\exp(x^2)}{x^{k+1}}\Big)$$

where $x$ is a positive real number. As far as I progressed, I obtained only

$$C(x) = \exp(x^2) + \frac{\exp(x^2)}{x} + O\Big(\frac{\exp(x^2)}{x}\Big).$$

I tried to use ideas from https://math.stackexchange.com/questions/484367/upper-bound-for-an-infinite-series-with-a-square-root?rq=1, https://math.stackexchange.com/questions/115410/whats-the-sum-of-sum-limits-k-1-infty-fractkkk, https://math.stackexchange.com/questions/378024/infinite-series-involving-sqrtn?noredirect=1&lq=1, but I was unable to make them work in my case.

Any suggestions would be greatly appreciated!

(If someone has a solid culture in this kind of things, is there are any specific names for $C(x) $ and $L(x) $ ?).

PS:

This question was asked on the math.SE but was closed as duplicate of https://math.stackexchange.com/questions/2117742/lim-x-rightarrow-infty-sqrtxe-x-left-sum-k%ef%bc%9d1-infty-fracxk/2123100#2123100. However, the latter question provides only the first term of the asymptotic expansion and does not address sufficiently the problem considered here.