关于级数的和函数一题求解

帮忙求解下列级数的和函数:
(1)$\sum\limits_{n=1}^{\infty} \cfrac{n^2+1}{2^n(n!)}x^n$; (2)$\sum\limits_{n=1}^{\infty} \cfrac{x^{4n}}{(4n)!}$.
已邀请:

Math001

赞同来自: 代数龙 poorich

第一题,注意到
$\sum\limits_{n=1}^{\infty} \cfrac{n^2+1}{2^n(n!)}x^n=\sum\limits_{n=1}^{\infty} \cfrac{n(n-1)+n+1}{n!}(\dfrac{x}{2})^n$

$=\sum\limits_{n=2}^{\infty} \cfrac{1}{(n-2)!}(\dfrac{x}{2})^n+\sum\limits_{n=1}^{\infty} \cfrac{1}{(n-1)!}(\dfrac{x}{2})^n+\sum\limits_{n=1}^{\infty} \cfrac{1}{n!}(\dfrac{x}{2})^n$

$=\sum\limits_{n=0}^{\infty} \cfrac{1}{n!}(\dfrac{x}{2})^{n+2}+\sum\limits_{n=0}^{\infty} \cfrac{1}{n!}(\dfrac{x}{2})^{n+1}+\sum\limits_{n=1}^{\infty} \cfrac{1}{n!}(\dfrac{x}{2})^n$

$=(\dfrac{x}{2})^2\sum\limits_{n=0}^{\infty}\cfrac{1}{n!}(\dfrac{x}{2})^{n}+(\dfrac{x}{2})\sum\limits_{n=0}^{\infty} \cfrac{1}{n!}(\dfrac{x}{2})^{n}+\sum\limits_{n=1}^{\infty} \cfrac{1}{n!}(\dfrac{x}{2})^n$

$=\dfrac{x^2}{4}e^{\frac{x}{2}}+\dfrac{x}{2}e^{\frac{x}{2}}+e^{\frac{x}{2}}-1$

第二题注意到,级数满足微分方程
$y^{(4)}=y+1$

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