Euler Sums系列(一)
\[\Large\sum_{n=1}^{\infty} \frac{H_{n}}{2^nn^4}\]
\(\Large\mathbf{Solution:}\)
Let
\[\mathcal{S}=\sum^\infty_{n=1}\frac{H_n}{n^42^n}\]
We first consider a slightly different yet related sum. The main idea is to solve this sum with two different methods, one of which involves the sum in question. This then allows us to determine the value of the desired sum.
\[\begin{align*}
&\sum^\infty_{n=1}\frac{(-1)^nH_n}{n^4}
=\frac{1}{6}\sum^\infty_{n=1}(-1)^{n-1}H_n\int^1_0x^{n-1}\ln^3{x}\ {\rm d}x
=\frac{1}{6}\int^1_0\frac{\ln^3{x}\ln(1+x)}{x(1+x)}{\rm d}x\\
=&\frac{1}{6}\int^1_0\frac{\ln^3{x}\ln(1+x)}{x}{\rm d}x-\frac{1}{6}\int^1_0\frac{\ln^3{x}\ln(1+x)}{1+x}{\rm d}x
=\frac{1}{6}\sum^\infty_{n=1}\frac{(-1)^{n-1}}{n}\int^1_0x^{n-1}\ln^3{x}\ {\rm d}x\\
-&\frac{1}{6}\int^2_1\frac{\ln{x}\ln^3(x-1)}{x}{\rm d}x
=\sum^\infty_{n=1}\frac{(-1)^{n}}{n^5}+\int^1_{\frac{1}{2}}\frac{\ln{x}\ln^3(1-x)}{6x}-\int^1_{\frac{1}{2}}\frac{\ln^2{x}\ln^2(1-x)}{2x}{\rm d}x\\+&\int^1_{\frac{1}{2}}\frac{\ln^3{x}\ln(1-x)}{2x}{\rm d}x-\int^1_{\frac{1}{2}}\frac{\ln^4{x}}{6x}{\rm d}x
=-\frac{15}{16}\zeta(5)+\mathcal{I}_1-\mathcal{I}_2+\mathcal{I}_3-\mathcal{I}_4
\end{align*}\]
Starting with the easiest integral,
\[\begin{align*}
\mathcal{I}_4=\frac{1}{30}\ln^5{2}
\end{align*}\]
For \(\mathcal{I}_3\),
\[\begin{align*}
\mathcal{I}_3
=&-\frac{1}{2}\sum^\infty_{n=1}\frac{1}{n}\int^1_{\frac{1}{2}}x^{n-1}\ln^3{x}\ {\rm d}x
=-\frac{1}{2}\sum^\infty_{n=1}\frac{1}{n}\frac{\partial^3}{\partial n^3}\left(\frac{1}{n}-\frac{1}{n2^n}\right)\\
=&\sum^\infty_{n=1}\left(\frac{3}{n^5}-\frac{3}{n^52^n}-\frac{3\ln{2}}{n^42^n}-\frac{3\ln^2{2}}{n^32^{n+1}}-\frac{\ln^3{2}}{n^22^{n+1}}\right)\\
=&3\zeta(5)-3{\rm Li}_5\left(\dfrac{1}{2}\right)-3{\rm Li}_4\left(\dfrac{1}{2}\right)\ln{2}-\frac{3}{2}\ln^2{2}\left(\frac{7}{8}\zeta(3)-\frac{\pi^2}{12}\ln{2}+\frac{1}{6}\ln^3{2}\right)\\&-\frac{1}{2}\ln^3{2}\left(\frac{\pi^2}{12}-\frac{1}{2}\ln^2{2}\right)\\
=&3\zeta(5)-3{\rm Li}_5\left(\dfrac{1}{2}\right)-3{\rm Li}_4\left(\dfrac{1}{2}\right)\ln{2}-\frac{21}{16}\zeta(3)\ln^2{2}+\frac{\pi^2}{12}\ln^3{2}
\end{align*}\]
For \(\mathcal{I}_2\),
\[\begin{align*}
\mathcal{I}_2
=&\frac{1}{6}\ln^5{2}+\frac{1}{3}\int^1_{\frac{1}{2}}\frac{\ln^3{x}\ln(1-x)}{1-x}{\rm d}x
=\frac{1}{6}\ln^5{2}-\frac{1}{3}\sum^\infty_{n=1}H_n\frac{\partial^3}{\partial n^3}\left(\frac{1}{n+1}-\frac{1}{(n+1)2^{n+1}}\right)\\
=&\frac{1}{6}\ln^5{2}+\sum^\infty_{n=1}\frac{2H_n}{(n+1)^4}-\sum^\infty_{n=1}\frac{2H_n}{(n+1)^42^{n+1}}-\sum^\infty_{n=1}\frac{2\ln{2}H_n}{(n+1)^32^{n+1}}\\
&-\sum^\infty_{n=1}\frac{\ln^2{2}H_n}{(n+1)^22^{n+1}}-\sum^\infty_{n=1}\frac{\ln^3{2}H_n}{3(n+1)2^{n+1}}\\
=&\frac{1}{6}\ln^5{2}+4\zeta(5)-\frac{\pi^2}{3}\zeta(3)-2\mathcal{S}+2{\rm Li}_5\left(\dfrac{1}{2}\right)-\frac{\pi^4}{360}\ln{2}+\frac{1}{4}\zeta(3)\ln^2{2}-\frac{1}{12}\ln^5{2}\\
&-\frac{1}{8}\zeta(3)\ln^2{2}+\frac{1}{6}\ln^5{2}-\frac{1}{6}\ln^5{2}\\
=&-2\mathcal{S}+2{\rm Li}_5\left(\dfrac{1}{2}\right)+4\zeta(5)-\frac{\pi^4}{360}\ln{2}+\frac{1}{8}\zeta(3)\ln^2{2}-\frac{\pi^2}{3}\zeta(3)+\frac{1}{12}\ln^5{2}
\end{align*}\]
For \(\mathcal{I}_1\),
\[\begin{align*}
\mathcal{I}_1
=&\frac{1}{6}\int^{\frac{1}{2}}_0\frac{\ln^3{x}\ln(1-x)}{1-x}{\rm d}x
=-\frac{1}{6}\sum^\infty_{n=1}H_n\frac{\partial^3}{\partial n^3}\left(\frac{1}{(n+1)2^{n+1}}\right)\\
=&\sum^\infty_{n=1}\frac{H_n}{(n+1)^42^{n+1}}+\sum^\infty_{n=1}\frac{\ln{2}H_n}{(n+1)^32^{n+1}}+\sum^\infty_{n=1}\frac{\ln^2{2}H_n}{2(n+1)^22^{n+1}}+\sum^\infty_{n=1}\frac{\ln^3{2}H_n}{6(n+1)2^{n+1}}\\
=&\mathcal{S}-{\rm Li}_5\left(\dfrac{1}{2}\right)+\frac{\pi^4}{720}\ln{2}-\frac{1}{16}\zeta(3)\ln^2{2}+\frac{1}{24}\ln^5{2}
\end{align*}\]
Combining these four integrals as \(\mathcal{I}_1-\mathcal{I}_2+\mathcal{I}_3-\mathcal{I}_4\) and \(\displaystyle -\dfrac{15}{16}\zeta(5)\) gives
\[\begin{align*}
\sum^\infty_{n=1}\frac{(-1)^nH_n}{n^4}
=&3\mathcal{S}-6{\rm Li}_5\left(\dfrac{1}{2}\right)-\frac{31}{16}\zeta(5)-3{\rm Li}_4\left(\dfrac{1}{2}\right)\ln{2}+\frac{\pi^4}{240}\ln{2}\\&-\frac{3}{2}\zeta(3)\ln^2{2}+\frac{\pi^2}{3}\zeta(3)+\frac{\pi^2}{12}\ln^3{2}-\frac{3}{40}\ln^5{2}
\end{align*}\]
But consider \(\displaystyle f(z)=\frac{\pi\csc(\pi z)(\gamma+\psi_0(-z))}{z^4}\). At the positive integers,
\[\sum^\infty_{n=1}{\rm Res}(f,n)=\sum^\infty_{n=1}\operatorname*{Res}_{z=n}\left[\frac{(-1)^n}{z^4(z-n)^2}+\frac{(-1)^nH_n}{z^4(z-n)}\right]=\sum^\infty_{n=1}\frac{(-1)^nH_n}{n^4}+\frac{15}{4}\zeta(5)\]
At \(z=0\),
\[\begin{align*}
{\rm Res}(f,0)
&=[z^3]\left(\frac{1}{z}+\frac{\pi^2}{6}z+\frac{7\pi^4}{360}z^3\right)\left(\frac{1}{z}-\frac{\pi^2}{6}z-\zeta(3)z^2-\frac{\pi^4}{90}z^3-\zeta(5)z^4\right)\\
&=-\zeta(5)-\frac{\pi^2}{6}\zeta(3)
\end{align*}\]
At the negative integers,
\[\begin{align*}
\sum^\infty_{n=1}{\rm Res}(f,-n)
&=\sum^\infty_{n=1}\frac{(-1)^nH_n}{n^4}+\frac{15}{16}\zeta(5)
\end{align*}\]
Since the sum of the residues is zero,
\[\sum^\infty_{n=1}\frac{(-1)^nH_n}{n^4}=-\frac{59}{32}\zeta(5)+\frac{\pi^2}{12}\zeta(3)\]
Hence,
\[\begin{align*}
-\frac{59}{32}\zeta(5)+\frac{\pi^2}{12}\zeta(3)
=&3\mathcal{S}-6{\rm Li}_5\left(\dfrac{1}{2}\right)-\frac{31}{16}\zeta(5)-3{\rm Li}_4\left(\dfrac{1}{2}\right)\ln{2}+\frac{\pi^4}{240}\ln{2}\\&-\frac{3}{2}\zeta(3)\ln^2{2}+\frac{\pi^2}{3}\zeta(3)+\frac{\pi^2}{12}\ln^3{2}-\frac{3}{40}\ln^5{2}
\end{align*}\]
This implies that
\[\boxed{\begin{align*}
{\sum^\infty_{n=1}\frac{H_n}{n^42^n}}
{=}&\color{blue}{2{\rm Li}_5\left(\dfrac{1}{2}\right)+\frac{1}{32}\zeta(5)+{\rm Li}_4\left(\dfrac{1}{2}\right)\ln{2}-\frac{\pi^4}{720}\ln{2}+\frac{1}{2}\zeta(3)\ln^2{2}}\\&\color{blue}{-\frac{\pi^2}{12}\zeta(3)-\frac{\pi^2}{36}\ln^3{2}+\frac{1}{40}\ln^5{2}}
\end{align*}}\]
最新文章
- WPF 自定义柱状图 BarChart
- ireport制作小技巧<;Reproduce>;
- 打造smali代码库辅助分析
- Jsoup使用随记
- NNVM打造模块化深度学习系统(转)
- HoloLens开发笔记之Gesture input手势输入
- Navicat链接Oracle提示ORA-12737
- if转换switch的小技巧
- JS之mouseover和mouseenter
- html+css学习总结
- Spring简单的文件配置
- java 方法的重载的语法规则
- Spring+SpringMvc+Mybatis 框架的搭建(一)
- Spring源码情操陶冶-AbstractApplicationContext#onRefresh
- Bin、App_Data等文件夹详述(转自http://blog.csdn.net/zzjiadw/article/details/6801506)
- Java语言
- Object类的方法
- linux-rhel7配置网卡bond双网卡主备模式
- IOS高级开发之多线程(五)NSOperation 2
- 记录1-更换mac pro内存,硬盘及恢复系统