Section 7.1

Definition 69 (Sequence)

A finite sequence is a function that has a set of natural numbers of the form \(\{1,2,3,...,n\}\) as its domain. An infinity sequence has the set of natural numbers as its domain.

The general term of the sequence is usually denoted \(a_{n}\).

Example 128

Write the first five terms of the sequence \(a_{n}=5n-2\).

Solution:

In this case the domain is \(\{1,2,3,4,5\}\) and

\[\begin{align*} a_{1} & =5(1)-2=3\\ a_{2} & =5(2)-2=8\\ a_{3} & =5(3)-2=13\\ a_{4} & =5(4)-2=18\\ a_{5} & =5(5)-2=23 \end{align*}\]

So the sequence \(\{a_{i}\}_{i=1}^{5}=\{3,8,13,18,23\}\).

Definition 70 (Convergence and Divergence)

If the terms of an infinite sequence get closer and closer to some real number, the sequence is said to be convergent and to converge to that real number.

A sequence that does not converge to any number is divergent.

The sequence \(a_{n}=\frac{1}{n}\) is a convergent sequence and converges to \(0\).

The sequence \(b_{n}=n^{2}\) is divergent and continue to get larger and larger without bound.

Example 129

If \(a_{1}=2\) and \(a_{n}=3\cdot a_{n-1}-1\) for \(n\ge1\), then find the first five terms of the sequence \(a_{n}\).

Solution:

We have

\[\begin{align*} a_{1} & =2\\ a_{2} & =3a_{2-1}-1\\ & =3a_{1}-1\\ & =3(2)-1\\ & =5\\ a_{3} & =3(5)-1\\ & =14\\ a_{4} & =3(14)-1\\ & =41\\ a_{5} & =3(41)-1\\ & =122 \end{align*}\]

Definition 71 (Series)

A finite series is an expression of the form

\[ S_{n}=a_{1}+a_{2}+\dots+a_{n}=\sum_{i=1}^{n}a_{i}. \]

An infinite series is an expression of the form

\[ S_{\infty}=a_{1}+a_{2}+\dots=\sum_{i=1}^{\infty}a_{i}. \]

We say the letter \(i\) is the index of the summation.

Example 130

Evaluate

\[ \sum_{i=1}^{5}(i^{2}+i+1). \]
Solution:

First we define \(a_{i}=i^{2}+i+1\) and we want to find the first \(5\) terms of the sequence.

\[\begin{align*} a_{1} & =(1)^{2}+1+1=3\\ a_{2} & =(2)^{2}+(2)+1=7\\ a_{3} & =(3)^{2}+(3)+1=13\\ a_{4} & =(4)^{2}+(4)+1=21\\ a_{5} & =(5)^{2}+(5)+1=31 \end{align*}\]

Next, we evaluate

\[\begin{align*} \sum_{i=1}^{5}(i^{2}+i+1) & =\sum_{i=1}^{5}a_{i}\\ & =a_{1}+a_{2}+a_{3}+a_{4}+a_{5}\\ & =3+7+13+21+31\\ & =75 \end{align*}\]

Therefore,

\[ \sum_{i=1}^{5}(i^{2}+i+1)=75. \]

Example 131

Evaluate \(\sum_{i=1}^{3}f(x_{i})\Delta x\) where \(f(x)=\frac{1}{x}\); \(x_{1}=1\) , \(x_{2}=3\), \(x_{3}=5\), and \(\Delta x=2\).

Solution:

First, we will define \(a_{i}=f(x_{i})\Delta x\) and we will find the first 3 terms of the sequence

\[\begin{align*} a_{1} & =f(x_{1})\Delta x\\ & =f(1)(2)\\ & =\frac{1}{1}(2)\\ & =2\\ a_{2} & =f(x_{2})\Delta x\\ & =f(3)(2)\\ & =\frac{1}{3}\cdot2\\ & =\frac{2}{3}\\ a_{3} & =f(x_{3})\Delta x\\ & =f(5)(2)\\ & =\frac{1}{5}\cdot2\\ & =\frac{2}{5}\\ \sum_{i=1}^{3}f(x_{i})\Delta x & =f(x_{1})\Delta x+f(x_{2})\Delta x+f(x_{3})\Delta x\\ & =2+\frac{2}{3}+\frac{2}{5}\\ & =\frac{46}{15} \end{align*}\]

Therefore, we have

\[ \sum_{i=1}^{3}f(x_{i})\Delta x=\frac{46}{15}. \]

Property 16 (Summation Properties)

If \(a_{1}\), \(a_{2}\), \(a_{3}\), …, \(a_{n}\) and \(b_{1}\), \(b_{2}\), \(b_{3}\), …, \(b_{n}\) are two sequences, and \(c\) is a constant, then for every positive integer \(n\), the following holds true.

  1. \(\sum_{i=1}^{n}c=cn\)

  2. \(\sum_{i=1}^{n}ca_{i}=c\sum_{i=1}^{n}a_{i}\)

  3. \(\sum_{i=1}^{n}(a_{i}+b_{i})=\sum_{i=1}^{n}a_{i}+\sum_{i=1}^{n}b_{i}\)

  4. \(\sum_{i=1}^{n}(a_{i}-b_{i})=\sum_{i=1}^{n}a_{i}-\sum_{i=1}^{n}b_{i}\)

  5. \(\sum_{i=1}^{n}i=\dfrac{n(n+1)}{2}\)

  6. \(\sum_{i=1}^{n}i^{2}=\dfrac{n(n+1)(2n+1)}{6}\)

  7. \(\sum_{i=1}^{n}i^{3}=\dfrac{n^{2}(n+1)^{2}}{4}\)

Example 132

Evaluate

\[ \sum_{i=1}^{10}(3i^{2}+5) \]
Solution:

Here we will use the properties of summation

\[\begin{align*} \sum_{i=1}^{10}(3i^{2}+5) & =\sum_{i=1}^{10}3i^{2}+\sum_{i=1}^{10}5\\ & =3\sum_{i=1}^{10}i^{2}+\sum_{i=1}^{10}5\\ & =3\cdot\frac{n(n+1)(2n+1)}{6}|_{n=10}+5n|_{n=10}\\ & =3\cdot\frac{(10)((10)+1)(2(10)+1)}{6}+5(10)\\ & =3\cdot385+50\\ & =1205 \end{align*}\]

Therefore,

\[ \sum_{i=1}^{10}(3i^{2}+5)=1205. \]