# 拉馬努金求和

## 求和法

{\displaystyle {\begin{aligned}{}&{\frac {1}{2}}f\left(0\right)+f\left(1\right)+\cdots +f\left(n-1\right)+{\frac {1}{2}}f\left(n\right)\\=&{\frac {1}{2}}\left[f\left(0\right)+f\left(n\right)\right]+\sum _{k=1}^{n-1}f\left(k\right)\\=&\int _{0}^{n}f(x)\,dx+\sum _{k=1}^{p}{\frac {B_{k+1}}{(k+1)!}}\left[f^{(k)}(n)-f^{(k)}(0)\right]+R_{p}\end{aligned}}}

${\displaystyle \sum _{k=1}^{x}f(k)=C+\int _{0}^{x}f(t)\,dt+{\frac {1}{2}}f(x)+\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(x)}$

${\displaystyle C(a)=\int _{0}^{a}f(t)\,dt-{\frac {1}{2}}f(0)-\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(0)}$

${\displaystyle C(a)=\int _{1}^{a}f(t)\,dt+{\frac {1}{2}}f(1)-\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(1)}$

C(0)因此被提議用作發散數列的和。在此建立了求和與積分之間的橋梁。

## 發散級數的和

${\displaystyle 1-1+1-1+\cdots ={\frac {1}{2}}\ (\Re )}$

${\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}}\ (\Re )}$

${\displaystyle 1+2^{2k}+3^{2k}+\cdots =0\ (\Re )}$

${\displaystyle 1+2^{2k-1}+3^{2k-1}+\cdots =-{\frac {B_{2k}}{2k}}\ (\Re )}$

${\displaystyle \sum _{n\geq 1}^{\Re }f(n)=\lim _{N\to \infty }\left[\sum _{n=1}^{N}f(n)-\int _{1}^{N}f(t)\,dt\right]}$

${\displaystyle \sum _{n\geq 1}^{\Re }{\frac {1}{n}}=\gamma }$

${\displaystyle {\begin{array}{l}\int \nolimits _{a}^{\infty }x^{m-s}dx={\frac {m-s}{2}}\int \nolimits _{a}^{\infty }x^{m-1-s}dx+\zeta (s-m)-\sum \limits _{i=1}^{a}i^{m-s}+a^{m-s}\\-\sum \limits _{r=1}^{\infty }{\frac {B_{2r}\Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)}}(m-2r+1-s)\int \nolimits _{a}^{\infty }x^{m-2r-s}dx\end{array}}}$

${\displaystyle \qquad \int _{a}^{\infty }dxx^{m-2r}=-{\frac {a^{m-2r+1}}{m-2r+1}}}$

${\displaystyle I(n,\,\Lambda )\,=\,\int _{0}^{\Lambda }dxx^{n}}$（參見：。）

## 參考文獻

1. ^ Bruce C. Berndt, Ramanujan's Notebooks 互联网档案馆存檔，存档日期2006-10-12., Ramanujan's Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.), (1939), pp. 133-149.
2. ^
3. ^ Infinite series are weird. [20 January 2014].
4. ^ Éric Delabaere, Ramanujan's Summation, Algorithms Seminar 2001–2002, F. Chyzak (ed.), INRIA, (2003), pp. 83–88.