# 38 Summation

## 38.1 Theorems of Summation

### 38.1.1 Theorem

If $$c$$ is a constant then $\sum\limits_{i=1}^{n}c = nc$

Proof:

$\sum\limits_{i=1}^{n}c = \underbrace{c+c+\cdots+c}_{n\ \rm terms} = nc$

### 38.1.2 Theorem

If $$a_1,a_2,\ldots,a_n$$ are real numbers and $$c$$ is a constant, then $\sum\limits_{i=1}^{n}ca_i = c\sum\limits_{i=1}^{n}a_i$

Proof:

\begin{aligned} \sum\limits_{i=1}^{n}ca_i &= ca_1 + ca_2 + \cdots + ca_n \\ &= c(a_1+a_2+\cdots+a_n) \\ &= c\sum\limits_{i=1}^{n}a_i \end{aligned}

### 38.1.3 Theorem

If $$a_1,_2,\ldots,a_n$$ are real numbers and $$b_1,b_2,\ldots,b_n$$ are real numbers, then $\sum\limits_{i=1}^{n}(a_i+b_i) = \sum\limits_{i=1}^{n}a_i + \sum\limits_{i=1}^{n}b_i$

Proof:

\begin{aligned} \sum\limits_{i=1}^{n}(a_i+b_i) &= a_1 + b_1 + a_2 + b_2 + \cdots + a_n + b_n \\ &= a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n \\ &= \sum\limits_{i=1}^{n}a_i + \sum\limits_{i=1}^{n}b_i \end{aligned}

### 38.1.4 Theorem

If $$a_i$$ and $$b_j$$ are real numbers for $$i=1,2,\ldots,n$$, $$j=1,2,\ldots,m$$, then then

$\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_i b_j = a_{+} b_{+}$

Proof:

\begin{aligned} \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_i b_j &= \sum\limits_{i=1}^{n}\bigg(a_i\sum\limits_{j=1}^{m}b_j\bigg) \\ &= \sum\limits_{i=1}^{n}a_i b_{+} \\ &= b_{+} \sum\limits_{i=1}^{n}a_i \\ &= a_{+} b_{+} \end{aligned}

### 38.1.5 Theorem

If $$a_i$$ is a real number for $$i=1,2,\ldots,n$$ and $$b$$ is a real number, then $\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_i b = m a_{+} b$

Proof:

\begin{aligned} \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_i b &= \sum\limits_{i=1}^{n}m a_i b \\ &= m b\sum\limits_{i=1}^{n}a_i \\ &= m a_{+} b \end{aligned}

### 38.1.6 Theorem

If $$a_j$$ is a real number for $$j=1,2,\ldots,m$$ and $$b$$ is a real number, then $\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_j b = n a_{+} b$

Proof:

\begin{aligned} \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_j b &= \sum\limits_{i=1}^{n}\bigg( b \sum\limits_{j=1}^{m} a_j \bigg) \\ &= \sum\limits_{i=1}^{n}a_{+}b \\ &= n a_{+} b \end{aligned}

### 38.1.7 Theorem

If $$a_i$$ and $$b_{ij}$$ are real numbers for $$i=1,2,\ldots,n$$, $$j=1,2,\ldots,m$$, then $\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_ib_{ij} = \sum\limits_{i=1}^{n}a_ib_{i+}$

Proof:

\begin{aligned} \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_ib_{ij} &= \sum\limits_{i=1}^{n}\bigg(a_i\sum\limits_{j=1}^{m}b_{ij}\bigg) \\ &= \sum\limits_{i=1}^{n}a_ib_{i+} \end{aligned}

### 38.1.8 Theorem

If $$a_j$$ and $$b_{ij}$$ are real numbers for $$i=1,2,\ldots,n$$, $$j=1,2,\ldots,m$$, then

$\ \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_jb_{ij} = \sum\limits_{i=1}^{n}a_jb_{+ j}$

Proof:

\begin{aligned} \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}a_jb_{ij} &= a_1b_{11}+a_2b_{12}+\cdots+a_mb_{1m} \\ & \ \ \ \ +a_1b_{21}+a_2b_{22}+\cdots+a_mb_{2m} \\ & \ \ \ \ \vdots \\ & \ \ \ \ +a_1b_{n1}+a_1b_{n1}+\cdots+a_1b_{nm} \\ &= a_1b_{11}+a_1b_{21}+\cdots+a_1b_{n1} \\ & \ \ \ \ +a_2b_{12}+a_2b_{22}+\cdots+a_2b_{n2} \\ & \ \ \ \ \vdots \\ & \ \ \ \ +a_mb_{1m}+a_mb_{2m}+\cdots+a_nb_{nm} \\ &= a_1(b_{11}+b_{21}+\cdots+b_{n1}) \\ & \ \ \ \ +a_2(b_{12}+b_{22}+\cdots+b_{n2}) \\ & \ \ \ \ \vdots \\ & \ \ \ \ +a_m(b_{1m}+b_{2m}+\cdots+b_{nm}) \\ &= a_1b_{+ 1}+a_2b_{+ 2}+\cdots+a_mb_{+ m} \\ &=\sum\limits_{j=1}^{m}a_jb_{+ j} \end{aligned}