# 求和符号

（重定向自總和

${\displaystyle 16=1+3+5+7}$

${\displaystyle \Sigma =x_{1}+x_{2}+\cdots +x_{n}}$

${\displaystyle \sum _{k=1}^{n}x_{k}}$

## 求和方法

1. 裂項法：利用${\displaystyle a_{k}=b_{k+1}-b_{k}}$求出${\displaystyle \sum _{k=m}^{n}a_{k}}$
2. 錯位相減法：透過兩個求和式的相減化簡求和數列的求和方法。
3. 倒序求和：對於有對稱中心的函數${\displaystyle f(x)+f(2a-x)=2b}$首尾求和[1][2]
4. 逐項求導：可從${\displaystyle \displaystyle \sum _{k=0}^{n}x^{k}={\frac {x^{n+1}-1}{x-1}}}$推導出${\displaystyle \displaystyle \sum _{k=0}^{n}k^{m}x^{k}}$[3]
5. 阿貝爾變換
${\displaystyle \sum _{i=1}^{n}a_{i}b_{i}=a_{1}(b_{1}-b_{2})+(a_{1}+a_{2})(b_{2}-b_{3})+\dots +(a_{1}+a_{2}+\dots +a_{n-1})(b_{n-1}-b_{n})+(a_{1}+a_{2}+\dots +a_{n})b_{n}}$

## 常見的總和公式

• 三角形數${\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}}}$
• ${\displaystyle \sum _{k=1}^{n}k^{0}=\sum _{k=1}^{n}1=n}$
• 連續正整數平方和：${\displaystyle \sum _{k=1}^{n}k^{2}={\frac {n(n+1)(2n+1)}{6}}}$
• 連續正整數立方和：${\displaystyle \sum _{k=1}^{n}k^{3}=\left[{\frac {n(n+1)}{2}}\right]^{2}}$
• 正方形數${\displaystyle \sum _{k=1}^{n}(2k-1)=n^{2}}$
• 調和級數${\displaystyle \sum _{n=1}^{k}\,{\frac {1}{n}}\;=\;\ln k+\gamma +\varepsilon _{k}}$

### 級數求和公式

${\displaystyle \sum _{i=0}^{n-1}(a_{1}+id)={\frac {n(a_{1}+a_{n})}{2}}={\frac {n[2a_{1}+(n-1)d]}{2}}=a_{1}C_{n}^{1}+dC_{n}^{2}}$
${\displaystyle \sum _{i=0}^{n-1}x^{i}={\frac {x^{n}-1}{x-1}}}$，若0 < |x| < 1，則${\displaystyle \sum _{i=0}^{\infty }x^{i}={\frac {1}{1-x}}}$
${\displaystyle \displaystyle \sum _{k=1}^{n}[a+(k-1)d]r^{k-1}=\left[{\frac {a+(n-1)d}{r-1}}-{\frac {d}{(r-1)^{2}}}\right]r^{n}-\left[{\frac {a-d}{r-1}}-{\frac {d}{(r-1)^{2}}}\right]}$

### 其他級數求和公式

#### ${\displaystyle \sum p(k)}$

${\displaystyle \sum p(k)}$是對一個多項式求和，自然數方冪和、等幂求和、等差數列求和都屬于對多項式求和。

• 帕斯卡矩陣形式
${\displaystyle \sum _{k=1}^{n}p(k)={\begin{pmatrix}C_{n}^{1}&C_{n}^{2}&\cdots &C_{n}^{m+1}\end{pmatrix}}{\begin{pmatrix}C_{0}^{0}&0&\cdots &0\\-C_{1}^{0}&C_{1}^{1}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\(-1)^{m}C_{m}^{0}&(-1)^{m-1}C_{m}^{1}&\cdots &C_{m}^{m}\\\end{pmatrix}}{\begin{pmatrix}p(1)\\p(2)\\\vdots \\p(m+1)\end{pmatrix}}}$[4]
• 差分變換形式
${\displaystyle p(k)=\sum _{j=1}^{m+1}C_{k-1}^{j-1}\Delta ^{j-1}p(1)}$
${\displaystyle \sum _{k=1}^{n}p(k)=\sum _{j=1}^{m+1}C_{n}^{j}\Delta ^{j-1}p(1)}$[5]

#### ${\displaystyle \sum u_{k}v_{k}x^{k}}$

${\displaystyle u_{k}=p(k)}$為多項式，${\displaystyle \sum _{l=0}^{\infty }v_{l}x^{l}}$易求高階導數時，${\displaystyle \sum _{k=0}^{\infty }u_{k}v_{k}x^{k}}$有封閉型和式

${\displaystyle \sum _{k=0}^{\infty }u_{k}v_{k}x^{k}=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}u_{0}x^{k}}{k!}}{\frac {d^{k}}{dx^{k}}}(\sum _{l=0}^{\infty }v_{l}x^{l})}$[6]
• ${\displaystyle u_{k}=p(k),v_{k}=1,x=q,\sum u_{k}v_{k}x^{k}=\sum p(k)q^{k}}$
有限和${\displaystyle \displaystyle \sum _{k=1}^{n}p(k)q^{k-1}}$有封閉型和式
當p為常數時，是對等比數列求和，當p為一次多項式時，是對差比數列求和。
${\displaystyle \displaystyle \sum _{k=1}^{n}p(k)q^{k-1}=f(n)q^{n}-f(0)}$
${\displaystyle f(n)={\frac {p(n)}{q-1}}+{\frac {1}{(q-1)^{2}}}\sum _{k=1}^{m}{\frac {(-1)^{k}q^{k-1}}{(q-1)^{k-1}}}\Delta ^{k}(p(n))={\frac {1}{q-1}}\sum _{k=0}^{m}({\frac {-q}{q-1}})^{k}\Delta ^{k}p(n+1)}$[7]
• ${\displaystyle u_{k}=p(k),v_{k}={\frac {1}{k!}},\sum {\frac {p(k)}{k!}}x^{k}}$
${\displaystyle \sum _{n=0}^{\infty }{\frac {p(n)}{n!}}x^{n}=e^{x}\sum _{k=0}^{m}{\frac {\Delta ^{k}p(0)}{k!}}x^{k}}$[8]

### 組合數求和公式

• ${\displaystyle \sum _{i=0}^{n}{\binom {n}{i}}=2^{n}}$
• ${\displaystyle \sum _{i=m}^{n}{\binom {k_{1}+i}{k_{2}}}={\binom {k_{1}+n+1}{k_{2}+1}}-{\binom {k_{1}+m}{k_{2}+1}}}$
• ${\displaystyle \sum _{i=m}^{n}{\binom {k_{1}+i}{k_{2}+i}}={\binom {k_{1}+n+1}{k_{2}+n}}-{\binom {k_{1}+m}{k_{2}+m-1}}}$

## 定積分判斷總和界限

${\displaystyle f(x)}$在[a,b]單調遞增時：

${\displaystyle f(a)+\int _{a}^{b}f(x)dx\leq \sum _{x=a}^{b}f(x)\leq f(b)+\int _{a}^{b}f(x)dx}$

${\displaystyle f(x)}$在[a,b]單調遞減時：

${\displaystyle f(b)+\int _{a}^{b}f(x)dx\leq \sum _{x=a}^{b}f(x)\leq f(a)+\int _{a}^{b}f(x)dx}$[9]

## 求和函数

${\displaystyle \sum _{i=1}^{n}i^{9}}$为例：

syms k n;symsum(k^9,k,1,n)

 In[1]:= Sum[i^9, {i, 1, n}]
Out[1]:= ${\displaystyle {\frac {1}{20}}n^{2}(n+1)^{2}\left(n^{2}+n-1\right)\left(2n^{4}+4n^{3}-n^{2}-3n+3\right)}$


## 参考资料

1. ^ 马志钢. 倒序求和几例. 中学生数学. 2006, (5).
2. ^ 郭子伟. 高中基础数列知识微型整理. 数学空间. 2011, (1): 第11页.
3. ^ 吴炜超. 数列{n^m.k^n}的求和方法. 数学空间. 2011, (7): 第38–39页.
4. ^ 黄嘉威. 方幂和及其推广和式. 数学学习与研究. 2016, (7).
5. ^ Károly Jordán. Calculus of Finite Differences.
6. ^ Murray Spiegel. Schaum's Outline of Calculus of Finite Differences and Difference Equations.
7. ^ 黄嘉威. 方幂和及其推广和式. 数学学习与研究. 2016, (7).
8. ^ 刘治国. 一类指数型幂级数的求和. 抚州师专学报. 1994, (01): 第65–66页.
9. ^ 吴炜超. 数列不等式的定积分解法. 数学空间. 2011, (5): 第23–26页.