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:

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