Chapter 2 Sequences and Series

2.1 Sequences

  • A Sequence is a list of numbers written in a definite order: \[a_{1}, a_{2}, a_{3}, a_{4}, ..., a_{n}\]
  • An element of a sequence is called a term . The number \(a_{1}\) is called the first term, \(a_{2}\) is the second term, and in general \(a_{n}\) is called the \(n^{th}\) term.
  • As can be seen in the above sequence, for every positive integer \(n\) there is a corresponding number \(a_{n}\).Therefore a sequence can be defined as a function whose domain is the set of positive integers.
  • But we usually write \(a_{n}\) instead of the function notation \(f(n)\) for the value of the function at the number \(n\).
  • When dealing with infinite sequences, each term \(a_{n}\) will have a successor \(a_{n+1}\)
  • Notation: The sequence \(\{a_{1}, a_{2}, a_{3},...\}\) is also denoted by \[\{a_{n}\}\] or \[\{a_{n}\}_{n=1}^{\infty}\]
  • Some sequences can be defined by giving a formula for the \(n\)th term.
  • In the following examples we give three descriptions of the sequences:
    1. by using the preceding notation
    2. by using the defining formula
    3. by writing out the terms of the sequence
  • Notice that \(n\) doesn’t have to start at 1.

  1. \(\{\frac{n}{n+1}\}_{n=1}^{\infty} \hspace{5em} a_{n} = \frac{n}{n+1}\hspace{5em}\{\frac{1}{2},\frac{2}{3}, \frac{3}{4}, \frac{4}{5},..., \frac{n}{n+1},... \}\)

  2. \(\{\frac{(-1)^n(n+1)}{3^{n}}\}_{n=1}^{\infty} \hspace{5em} a_{n} = \frac{(-1)^n(n+1)}{3^{n}}\hspace{5em}\{-\frac{2}{3},\frac{3}{9}, -\frac{4}{27}, \frac{5}{81},..., \frac{(-1)^n(n+1)}{3^{n}},... \}\)

  3. \(\{\sqrt{n-3}\}_{n=3}^{\infty} \hspace{5em} a_{n} = \sqrt{n-3}, n\geq3\hspace{5em}\{0,1,\sqrt{2}, \sqrt{3},..., \sqrt{n-3},...\}\)

  4. \(\{cos\frac{n\pi}{6}\}_{n=0}^{\infty} \hspace{5em} a_{n} = cos\frac{n\pi}{6}, n\geq 0 \hspace{5em}\{1, \frac{\sqrt{3}}{2},\frac{1}{2}, 0,...,cos\frac{n\pi}{6},...\}\)

2.2 Series

  • A series is the sum of a number of terms of a sequence.
  • When writing series, the shorthand \(\sum\) notation is used to represent the sum of a number of terms having a common form.
  • The series \(f(1)+ f(2)+\dots+ f(n−1)+ f(n)\) can be written as \[\sum_{r=1}^nf(r).\]

2.2.1 Arithmetic sequences and series

  • A sequence in which each term after the first term is obtained from the preceding term by adding a fixed number (Common difference), is called an arithmetic sequence or Arithmetic Progression.

  • The sequence defined by \[u_1=a \text{ and } u_n=u_{n-1} + d \text{ for } n\geq2\] gives \[a,a+d, a+2d,\dots\]
  • The \(n\)th term (i.e. the solution) is given by \(u_n = a + (n −1)d\).

  • This is the arithmetic sequence with first term \(a\) and common difference \(d\)

  • The arithmetic series with \(n\) terms \[a+(a+d)+(a+2d)+\dots+ [a+(n-1)d]\] has sum \[S_n = \frac{n}{2}(\text{first term} + \text{ last term)}\] \[S_n = \frac{n}{2}(2a+(n-1)d)\]

2.2.2 Geometric sequences and series

  • The sequence defined by \[u_1=a \text{ and } u_n=ru_{n-1} \text{ for } n\geq2\] gives \[a,ar, ar^2,\dots\]
  • The \(n\)th term is given by \(u_n = ar^{n −1}\).

  • This is the geometric sequence with first term \(a\) and common ratio \(r\).

  • The geometric series with \(n\) terms \[a+ar+ar^2+\dots+ ar^{n-1}\] has sum \[S_n = \frac{a(1-r^n)}{1-r} \text{ or } \frac{a(r^n-1)}{r-1} \text{ for } r\neq1\]

Important results relating to the \(\sum\) notation

  1. \[\sum_{r=1}^n\{f(r)+ g(r)\} = \{f(1)+ g(1)\}+ \{f(2)+ g(2)\}+\dots + \{f(n)+ g(n)\}\] \[= \{f(1)+ f(2)+ \dots+ f(n)\}+ \{g(1)+ g(2)+\dots +g(n)\}\] \[= \sum_{r=1}^nf(r)+ \sum_{r=1}^ng(r)\]

  2. \[\sum_{r=1}^naf(r)= af(1)+ af(2)+ \dots+ af(n)\] where \(a\) is a constant. \[=a\{f(1)+ f(2)+ \dots+ f(n)\}\] \[=a\sum_{r=1}^nf(r)\]

  1. \(\sum_{r=1}^nr=\frac{n(n+1)}{2}\)

  2. \(\sum_{r=1}^nr^2=\frac{n}{6}(n+1)(2n+1)\)

  3. \(\sum_{r=1}^nr^3=\frac{n^2}{4}(n+1)^2\) or \(\left[\frac{n(n+1)}{2}\right]^2\)

  4. \(\sum_{r=1}^n1=(1+1+\dots+1) = n\)

2.2.3 Methods of proof

  1. Mathematical induction
  2. The difference method

2.2.3.1 Mathematical induction

  • Let \(n\) be a natural number. Then the aim is to show that some statement \(P(n)\) involving \(n\) is true for any \(n\).

  • The following steps are used in Mathematical induction

    1. Let \(P(n)\) be a statement
    2. Show that the statement is true for \(P(1)\) and \(P(2)\). (i.e. \(P(n)\) is true for \(n=1\) and \(n=2.\))
    3. Assume that \(P(k)\) is true (i.e. \(P(n)\) is true for \(n=k\)).
    4. Show that \(P(k+1)\) follows from \(P(k).\)

2.2.3.2 The difference method

  • The process of proof by induction is a powerful mathematical tool. However it has the disadvantage that, in order to employ the method it requires the answer.
  • There are, however, direct methods of proof such as the method of differences, or the difference method.
  • The difference method can be summarised as follows,

\[\sum_{r=1}^n \left\{f(r)-f(r-1) \right\} = f(n) - f(0)\]

where \(f\) is any function suitably defined on the non-negative integers.

  • This is also know as the fundamental theorem of summation:

2.3 Infinite Sequences

  • When dealing with infinite sequences, each term \(a_{n}\) will have a successor \(a_{n+1}\)

Definition: Convergence and Divergence

A sequence \(\{a_n\}\) has the limit \(L\) and we write

\[\lim_{n\rightarrow \infty}a_n=L \text{ or } a_n \rightarrow L \text{ as } n\rightarrow \infty\]

if we can make the term \(a_n\) as close to L as \(n\) becomes sufficiently large.

If \(\lim_{n\rightarrow \infty}a_n\) exists, we say the sequence converges (or is convergent).

If \(\lim_{n\rightarrow \infty}a_n\) does not exist, we say the sequence diverges (or is divergent).

2.3.1 Limit Laws for Sequences

If \(\{a_n\}\) and \(\{b_n\}\) are convergent sequences and \(c\) is a constant, then

  1. \(\lim_{n \to \infty}(a_n+ b_n)= \lim_{n \to \infty}a_n+ \lim_{n \to \infty}b_n\)

  2. \(\lim_{n \to \infty}(a_n- b_n)= \lim_{n \to \infty}a_n- \lim_{n \to \infty}b_n\)

  3. \(\lim_{n \to \infty}ca_n= c \;\lim_{n \to \infty}a_n\)

  4. \(\lim_{n \to \infty}c= c\)

  5. \(\lim_{n \to \infty}(a_nb_n)= \lim_{n \to \infty}a_n. \lim_{n \to \infty}b_n\)

  6. \(\lim_{n \to \infty}\frac{a_n}{b_n}= \frac{\lim_{n \to \infty}a_n} {\lim_{n \to \infty}b_n} \text{ if } \lim_{n \to \infty}{b_n} \neq 0\)

  7. \(\lim_{n \to \infty}(a_n)^p= [\lim_{n \to \infty}a_n]^p \text{ if } p>0 \text{ and } a_n>0\)

Squeeze Theorem for Sequences

If \(a_n \leq b_n \leq c_n\) for \(n\geq n_0\) and \(\lim_{n \to \infty}a_n= \lim_{n \to \infty}c_n= L,\) then \(\lim_{n \to \infty}b_n=L\)

Theorem:

\[\text{If }\;\; \lim_{n \to \infty}|a_n|=0, \text{ then } \;\;\lim_{n \to \infty}a_n=0.\]

Theorem

The sequence \(\{r^n\}\) is convergent if \(-1 <r\leq1\) and divergent for all other values of \(r\).

\[\begin{equation} \lim_{n \to \infty}r^n= \begin{cases} 0 & \text{if } -1<r<1\\ 1 & \text{if } r=1 \end{cases} \end{equation}\]

Definition

A sequence \(\{a_n\}\) is called increasing if \(a_n<a_{n+1}\) for all \(n \geq 1,\) that is, \(a_1<a_2<a_3<\dots.\) It is called decreasing if \(a_n>a_{n+1}\) for all \(n\geq 1.\) It is called monotonic if it is either increasing or decreasing.

Definition

  • If there exists a number \(m\) such that \(m \leq a_n\) for every \(n\) we say the sequence is bounded below. The number \(m\) is sometimes called a lower bound for the sequence.

  • If there exists a number \(M\) such that \(a_n \leq M\) for every \(n\) we say the sequence is bounded above. The number \(M\) is sometimes called an upper bound for the sequence.

  • If the sequence is both bounded below and bounded above we call the sequence bounded.

2.4 Infinite Series

  • Consider the infinite sequence \(\{a_n\}_{n=1}^\infty\).
  • Consider the partial sums

\[\begin{align*} s_1=a_1 \\ s_2=a_1+a_2 \\ s_3=a_1+a_2+a_3 \\ s_4=a_1+a_2+a_3+a_4 \\ &\vdots\\ s_n=a_1+a_2+a_3+a_4+\dots +a_n =\sum_{i=1}^na_i \end{align*}\]

  • These partial sums form a new sequence \(\{s_n\}_{n=1}^\infty,\) which may or may not have a limit.

  • Consider the limit of the sequence of partial sums, \(\{s_n\}_{n=1}^\infty.\)

\[\lim_{n\to\infty}s_n= \lim_{n\to\infty}\sum_{i=1}^na_i=\sum_{i=1}^\infty a_i\]

  • This \(\sum_{i=1}^\infty a_i\) (\(=a_1+a_2+a_3+a_4+\dots +a_n \dots\)) is called an infinite series

Definition

Consider the series \(\sum_{n=1}^\infty a_n= =a_1+a_2+a_3+\dots.\) Let \(s_n\) denote its \(n\)th partial sum:

\[s_n= \sum_{i=1}^n a_i= a_1+a_2+\dots +a_n\]

If the sequence \(\{s_n\}\) is convergent and \(\lim_{n\to \infty}s_n=s\) exist as a real number, then the series \(\sum_{i=1}^\infty a_i\) is called convergent and we write,

\(a_1+a_2+\dots +a_n+\dots=s\) or \(\sum_{n=1}^\infty a_n=s\)

The number \(s\) is called the sum of the series. If the sequence of partial sums is divergent then the infinite series is also called divergent.

Geometric Series

  • An important example of an infinite series is the geometric series

\[\sum_{n=1}^\infty ar^{n-1}=a+ar+ar^2+\dots\] is convergent if \(|r|<1\) and its sum is \[\sum_{n=1}^\infty ar^{n-1}= \frac{a}{1-r}\qquad\qquad|r|<1\]

If \(|r|\geq1,\) the geometric series is divergent.

Theorem

If the series \(\sum_{n=1}^\infty a_n\) is convergent, then \(\lim_{n\to \infty}a_n=0.\)

Proof

Note 1

  • Consider any series \(\sum a_n.\)

  • This associates two sequences

    • the sequence \(\{s_n\}\) of its partial sums
    • the sequence \(\{a_n\}\) of its terms.

  • If \(\sum a_n\) is convergent, then

    • the limit of the sequence \(\{s_n\}\) is \(s\) (the sum of the series)
    • the limit of the sequence \(\{a_n\}\) is 0.

Note 2 - The converse of the above theorem is not true in general.

The Test for Divergence

If \(\lim_{n\to \infty}a_n\) does not exist or if \(\lim_{n\to \infty}a_n \neq0\), then the series \(\sum_{n=1}^\infty a_n\) is divergent.

Theorem

If \(\sum a_n\) and \(\sum b_n\) are convergent series, then

  1. \(\sum_{n=1}^\infty ca_n= c\sum_{n=1}^\infty a_n \;\;\) \(c\) is a constant.

  2. \(\sum_{n=1}^\infty (a_n+b_n)= \sum_{n=1}^\infty a_n+ \sum_{n=1}^\infty b_n\)

  3. \(\sum_{n=1}^\infty (a_n-b_n)= \sum_{n=1}^\infty a_n- \sum_{n=1}^\infty b_n\)

Reading:

  • Stewart, J., Clegg, D. K., & Watson, S. (2020). ‘Infinite Sequences and Series’, Calculus: early transcendentals. Cengage Learning.

  • Acharjya, D. P. (2009). Fundamental approach to discrete mathematics. New Age International.

Tutorial

  1. Find a formula for the general term \(a_{n}\) of the sequence, assuming that the pattern of the first few terms continues.
  1. \(\left\{\frac{3}{5},-\frac{4}{25}, \frac{5}{125}, -\frac{6}{625}, \frac{7}{3125},...\right\}\)
  2. \(\{1,1,2,3,5,8,13,21,\dots\}\)
  3. \(\left\{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots \right\}\)
  4. \(\left\{2,7,12,17, \dots \right\}\)
  5. \(\left\{1,-\frac{2}{3}, \frac{4}{9} , -\frac{8}{27}, \dots \right\}\)
  1. Show that \[\sum_{r=1}^n(6r^2+4r-1) = n(n+2)(2n+1)\]
  1. Show by method of induction
  1. \(\sum_{r=1}^nr=\frac{n(n+1)}{2}\)
  2. \(\sum_{r=1}^nr^2=\frac{n}{6}(n+1)(2n+1)\)
  3. \(\sum_{r=1}^nr^3=\left[\frac{n(n+1)}{2}\right]^2\)
  4. \(\sum_{r=1}^n\frac{1}{r(r+1)}=\frac{n}{n+1}\)

  1. By considering \(n^3 − (n − 1)^3\) and similar expressions, find the formula for \(\sum_{r=1}^nr^2\) in terms of \(n\), assuming the results for \(\sum_{r=1}^nr\)

  2. By considering \(n^5 − (n − 1)^5\) and similar expressions, find the formula for \(\sum_{r=1}^nr^4\) in terms of \(n\), assuming the results for \(\sum_{r=1}^nr\), \(\sum_{r=1}^nr^2\) and \(\sum_{r=1}^nr^3.\)

  1. Determine whether the sequence \(a_n= (-1)^n\) is convergent or divergent.
  1. Determine whether the sequence converges or diverges. If it converges, find the limit
  1. \(a_n= n(n-1)\)
  2. \(a_n= \frac{n}{n+1}\)
  3. \(a_n = \frac{(-1)^n}{n}\)
  4. \(a_n = \frac{4n^2+2}{n^2+n}\)
  5. \(a_n= \frac{2^n}{3^{n+1}}\)
  6. \(a_n= \frac{(-1)^{n-1}n}{n^2+1}\)
  1. Discuss the convergence of the sequence \(a_n=n!/n^n,\) where \(n!= 1\times2\times\dots\times n.\)
  1. Determine if the following sequences are monotonic and/ or bounded
  1. \(\{-n^2\}_{n=0}^\infty\)
  2. \(\{(-1)^{n+1}\}_{n=1}^\infty\)
  3. \(\left\{ \frac{2}{n^2}\right\}_{n=5}^\infty\)
  1. Determine if the following series converges or diverges. If it converges determine its value.
  1. \(\sum_{n=1}^\infty n\)
  2. \(\sum_{n=1}^\infty (-1)^n\)
  3. \(\sum_{n=1}^\infty \frac{1}{3^{n-1}}\)
  4. \(5-\frac{10}{3}+\frac{20}{9}- \frac{40}{27}+\dots\)
  5. \(\sum_{n=1}^\infty 2^{2n}3^{1-n}\)
  6. \(\sum_{n=1}^\infty x^n,\) where \(|x| <1\)
  7. \(\sum_{n=1}^\infty\frac{1}{n(n+1)}\)
  1. Show that harmonic series \[\sum_{n=1}^\infty \frac{1}{n} = 1+\frac{1}{2}+ \frac{1}{3} + \frac{1}{4}+\dots\] is divergent.

  1. Show that the series \(\sum_{n=1}^\infty\frac{n^2}{5n^2+4}\) diverges.

  2. Determine whether the series is convergent or divergent. If it is convergent, find its sum.

  1. \(\sum_{n=1}^\infty\left(\frac{3}{n(n+1)+ }+\frac{1}{2^n}\right)\)
  2. \(\sum_{n=1}^\infty\left(\frac{3}{n(n+3)+ }+\frac{5}{4^n}\right)\)
  3. \(\sum_{n=1}^\infty \frac{1}{n^2+3n+2}\)
  4. \(\sum_{n=1}^\infty \frac{1}{n^2+4n+3}\)
  5. \(\sum_{n=1}^\infty 9^{-n+2}4^{n+1}\)
  6. \(\sum_{n=1}^\infty \frac{(-4)^{3n}}{5^{n-1}}\)
  7. \(\sum_{n=1}^\infty \left(\frac{1}{n^2+4n+3}- 9^{-n+2}4^{n+1}\right)\)