How to compute $\sum_{n=0}^{\infty}\frac{(2n+1)}{n!}x^{2n+1}$? I don't know how to compute $\sum_{n=0}^{\infty}\frac{(2n+1)}{n!}x^{2n+1}$,appreciate any help!
Is there any general rule for solving such problems?
 A: $$e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$$
$$xe^{x^2}=\sum_{n=0}^\infty\frac{x^{2n+1}}{n!}$$
$$\frac{d}{dx}xe^{x^2}=\sum_{n=0}^\infty\frac{(2n+1)x^{2n}}{n!}$$
$$xe^{x^2}(2x^2+1)=\sum_{n=0}^\infty\frac{(2n+1)x^{2n+1}}{n!}$$
A: Hints:
I don't think there's a general rule but knowing some stuff can help:
$$f(x)=\sum_{n=0}^\infty\frac{2n+1}{n!}x^{2n}\implies\int f(x)\,dx=\sum_{n=0}^\infty\frac{x^{2n+1}}{n!}=x\sum_{n=0}^\infty\frac{(x^2)^n}{n!}=xe^{x^2}\ldots$$
Note that your series is $\,xf(x)\,\ldots$ , and also check where can you perform the above integration memberwise on that sum
A: An alternate approach, without taking derivatives or anti-derivatives. Let $$g(z)=\sum_{n=0}^\infty \frac{2n+1}{n!}z^n$$
Then the function for the series you asked about is $xg(x^2)$.
But $$\sum \frac{2n+1}{n!}z^n = \sum \frac{2}{(n-1)!}z^n + \sum \frac{1}{n!}z^n = 2ze^z + e^z = (2z+1)e^z$$
So $xg(x^2) = x(2x^2+1)e^{x^2}$.
For a general polynomial $p(n)$ we can write $p$ as a linear combination of falling factorials, $(n)_k=n(n-1)\dots(n-k+1)$.  So $p(n)=\sum_k a_k(n)_k$ for some finite sequence of values $a_k$ and then:
$$\sum_n \frac{p(n)}{n!} z^n = e^z \sum_k a_kz^k$$
For example, if $p(n)=n^2 = n(n-1) + n = (n)_2+(n)_1$, then $$\sum_n \frac{n^2}{n!}z^n = e^z(z^2+z)$$
Or $n^3 = (n)_3 + 3(n)_2 + (n)_1$ so:
$$\sum_n \frac{n^3}{n!}z^n = e^z(z^3+3z^2+z)$$
A: $$\text{As }  e^x=\sum_{0\le r<\infty}\frac{x^r}{r!} $$
 $$\text{and }\frac{(2n+1)}{n!}x^{2n+1}=2x^3 \cdot \frac{(x^2)^{n-1}}{(n-1)!}+x\cdot \frac{(x^2)^n}{n!}$$
$$\text{So,} \sum_{n=0}^{\infty}\frac{(2n+1)}{n!}x^{2n+1}$$
$$=2x^3 \cdot \sum_{n=0}^{\infty}\frac{(x^2)^{n-1}}{(n-1)!}+x\cdot \sum_{n=0}^{\infty}\frac{(x^2)^n}{n!}$$
$$=2x^3 \cdot \sum_{m=0}^{\infty}\frac{(x^2)^m}{m!}+x\cdot \sum_{n=0}^{\infty}\frac{(x^2)^n}{n!}\text{ putting }  n-1=m  \text{ and using} \frac1{(-1)!}=0 $$
$$=2x^3 \cdot e^{x^2}+x\cdot e^{x^2}$$
