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Number and Algebra

Sequences, Series and Proof

Sequences, Series and the \Sigma Notation

Sequences
  • A sequence is a list of numbers written down in a definite order, following a specific rule. Each of the numbers in this list is referred to as a term.

  • A sequence is denoted by \{u_r\} where r can take values 1,2,3, \dots

  • The r\text{th} term of a sequence is denoted by u_r.

  • Sequences may be finite or infinite.

  • Ellipsis (\dots) at the end of a sequence indicates an infinite sequence.

  • The sequence 7,9,11,13 is a finite sequence and can be written as \{u_r\}=\{2r+5\}, where r\in \mathbb{Z^+}, r\leq 4.

  • The infinite sequence 1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},... can be rewritten as \{u_r\}=\{\frac{1}{r^2}\}.

Series
  • The terms of a sequence considered as a sum, for instance 7+5+3+1 is called a series.

  • Like sequences, series can be finite or infinite.

  • The (infinite) set of even numbers can be written as \{2,4,6,8,\ldots, 2r, \ldots\}. The general term here is 2r where r\in \mathbb{Z^+}.

The \Sigma Notation
  • A series can be written compactly using sigma notation.

  • The general term written in terms of r and the range of values r can take are required to write a series using this notation.

  • For instance: the series 1^2 + 2^2 + \dots has the general term r^2 and r can take values 1,2,3,\dots so, we write \displaystyle \sum_{r=1}^{\infty} r^2 (read as "The sum of r^2, from r=1 to r=\infty).

  • A sum given in sigma notation can also be expanded into individual terms.

  • For example:

    \begin{aligned}\sum_{r=3}^{6} r(r+3)&=[3(3+3)]+[4(4+3)]+[5(5+3)]+[6(6+3)] \\ &=3\times 6 + 4\times 7+5\times 8+6\times 9.\end{aligned}

  • A sequence is said to be an arithmetic sequence or arithmetic progression if the difference between a term and the previous one is constant, called the common difference.

  • The nth term of an arithmetic sequence is obtained by adding \mathbf{n-1} common differences to the first term.

  • Thus, an arithmetic sequence with first term u_1 and common difference d has the general term \displaystyle u_n=u_1+(n-1)d.

  • The formula for the sum of a finite arithmetic progression is \displaystyle S_n=\frac{n}{2}(u_1+u_n)=\frac{n}{2}[2u_1+(n-1)d].

  • It can be derived as follows: \begin{array}{ccccccccccccc} S_n&=&u_1&+&u_1+d&+&u_1+2d&+&\ldots&+&u_n-d&+&u_n \\ S_n&=&u_n&+&u_n-d&+&u_n-2d&+&\ldots&+&u_1+d&+&u_1 \\\hline 2S_n&=&u_1+u_n&+&u_1+u_n&+&u_1+u_n&+&\ldots&+&u_1+u_n&+&u_1+u_n\end{array}

    Since there are n terms on the right-hand side, it follows that:

    \begin{aligned} 2S_n&=n(u_1+u_n) \\\bm{S_n}&\bm{=}\bm{\frac{n}{2}\left(u_1+u_n\right)}.\end{aligned}

  • A sequence is said to be a geometric sequence or geometric progression if the ratio of a term to the previous one is constant.

  • The constant ratio is referred to as common ratio and is denoted by r.

  • The \boldsymbol{n}th term of a geometric sequence is obtained by multiplying the first term by the \bm{(n-1)}th power of the common ratio.

  • Thus, a geometric difference with first term u_1 and common ratio r has the general term u_n=u_1r^{n-1}, where r\neq 0,1,-1 and u_1\neq1.

  • The formula for the sum of a finite geometric progression is \displaystyle S_n=\frac{u_1(1-r^n)}{1-r},\quad r\neq1.

  • The derivation is as shown (note the cancellations):

    \begin{array}{ccccccccccccccc} \phantom{aaaaaaa}S_n&=&&&u_1&+&u_1r&+&u_1r^2&+&\ldots&+&u_1r^{n-1}\\ - \\ \phantom{aaaaaaa}rS_n&=&&&&& u_{1}r&+&u_{1}r^2&+&\ldots&+&u_1r^{n-1}&+&u_1r^n \\ \hline (1-r) S_n&=&&&u_1-u_1r^n& \end{array}

    \displaystyle \hspace{0.95cm} \implies \large S_n=\frac{u_1(1-r^n)}{1-r}, \quad (r\neq 1)

  • If \lvert r \rvert <1, then for large values of n, r^n approaches zero and the formula becomes \displaystyle S_n=\frac{u_1}{1-r}.

Proof

  • A proof in mathematics is an argument consisting of a logical set of steps that validates the truth of a general statement beyond any doubt.

Types of proofs
  • A direct proof is a method of proof that involves constructing a series of reasoned connected established facts.

  • To write a direct proof, you need to:

    • identify the given mathematical statement

    • use axioms, theorems, etc. to make deductions that prove the conclusion of a given statement.

\large \text{Example}

We can show that two numbers always sum up to an even number using direct proof.

Let m and n be two odd positive integers.

Then m=2p+1 and n=2q+1, where p,q \in \mathbb{Z}^+.

Therefore, m+n=2p+2q+2=2(p+q+1).

Since p+q+1 \in \mathbb{Z^+}, m+n is even.

  • A statement may not always be easily proved directly. In which case you need to employ a different type of proof.

  • Statements can also be proved using contradiction. For a proof by contradiction, the following steps are involved:

    • identify the implication of the given statement

    • assume that the implication is false

    • use axioms, theorems, etc. to produce a contradiction

    • conclude that the original statement is true.

\large \text{Example}

The number \sqrt{2} can be shown to be irrational using contradiction.

Assume (in search of a contradiction) that \sqrt{2} is rational that is \sqrt{2}=\frac{m}{n} where m,n \in \mathbb{Z}, m,n have no common factors and n\neq 0.

Then 2=\frac{m^2}{n^2}; so m^2=2n^2.

This means m^2 is an even integer since n^2 is an integer.

It follows that m is even. So m=2k and m^2=4k^2.

Therefore, m^2=4k^2=2n^2 which implies that n^2=2k^2. This means that n^2 is also an even integer.

Since m and n are even integers, they have a common factor of 2, which contradicts the assumption that m, n have no common factors.

  • An example which contradicts a given statement can be used to justify that a statement or conjecture is false.

  • Finding such an example is the basis of a proof known as counterclaim or counterexample.

\large \text{Example}

Consider the statement if n\in \mathbb{Z}, then n^2+1 is prime. For n=3,\ n^2+1=10 which is not a prime number. Thus, n=3 is a counterexample.

  • Yet another type of proof is proof by induction which makes use of the principle of induction.

  • Mathematical induction is a technique for proving that a statement about an integer $n$ is true for every integer n greater than or equal to some starting integer n_0.

  • Let P(n) be a statement pertaining each positive integer n. If

    • P(1) is true and

    • the implication: “If P(k) then P(k+1)” is true for every positive integer k,

      then P(n) is true for every positive integer n. This is the principle of mathematical induction.

Number and Algebra

Sequences, Series and Proof

Sequences, Series and the \Sigma Notation

Sequences
  • A sequence is a list of numbers written down in a definite order, following a specific rule. Each of the numbers in this list is referred to as a term.

  • A sequence is denoted by \{u_r\} where r can take values 1,2,3, \dots

  • The r\text{th} term of a sequence is denoted by u_r.

  • Sequences may be finite or infinite.

  • Ellipsis (\dots) at the end of a sequence indicates an infinite sequence.

  • The sequence 7,9,11,13 is a finite sequence and can be written as \{u_r\}=\{2r+5\}, where r\in \mathbb{Z^+}, r\leq 4.

  • The infinite sequence 1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},... can be rewritten as \{u_r\}=\{\frac{1}{r^2}\}.

Series
  • The terms of a sequence considered as a sum, for instance 7+5+3+1 is called a series.

  • Like sequences, series can be finite or infinite.

  • The (infinite) set of even numbers can be written as \{2,4,6,8,\ldots, 2r, \ldots\}. The general term here is 2r where r\in \mathbb{Z^+}.

The \Sigma Notation
  • A series can be written compactly using sigma notation.

  • The general term written in terms of r and the range of values r can take are required to write a series using this notation.

  • For instance: the series 1^2 + 2^2 + \dots has the general term r^2 and r can take values 1,2,3,\dots so, we write \displaystyle \sum_{r=1}^{\infty} r^2 (read as "The sum of r^2, from r=1 to r=\infty).

  • A sum given in sigma notation can also be expanded into individual terms.

  • For example:

    \begin{aligned}\sum_{r=3}^{6} r(r+3)&=[3(3+3)]+[4(4+3)]+[5(5+3)]+[6(6+3)] \\ &=3\times 6 + 4\times 7+5\times 8+6\times 9.\end{aligned}

  • A sequence is said to be an arithmetic sequence or arithmetic progression if the difference between a term and the previous one is constant, called the common difference.

  • The nth term of an arithmetic sequence is obtained by adding \mathbf{n-1} common differences to the first term.

  • Thus, an arithmetic sequence with first term u_1 and common difference d has the general term \displaystyle u_n=u_1+(n-1)d.

  • The formula for the sum of a finite arithmetic progression is \displaystyle S_n=\frac{n}{2}(u_1+u_n)=\frac{n}{2}[2u_1+(n-1)d].

  • It can be derived as follows: \begin{array}{ccccccccccccc} S_n&=&u_1&+&u_1+d&+&u_1+2d&+&\ldots&+&u_n-d&+&u_n \\ S_n&=&u_n&+&u_n-d&+&u_n-2d&+&\ldots&+&u_1+d&+&u_1 \\\hline 2S_n&=&u_1+u_n&+&u_1+u_n&+&u_1+u_n&+&\ldots&+&u_1+u_n&+&u_1+u_n\end{array}

    Since there are n terms on the right-hand side, it follows that:

    \begin{aligned} 2S_n&=n(u_1+u_n) \\\bm{S_n}&\bm{=}\bm{\frac{n}{2}\left(u_1+u_n\right)}.\end{aligned}

  • A sequence is said to be a geometric sequence or geometric progression if the ratio of a term to the previous one is constant.

  • The constant ratio is referred to as common ratio and is denoted by r.

  • The \boldsymbol{n}th term of a geometric sequence is obtained by multiplying the first term by the \bm{(n-1)}th power of the common ratio.

  • Thus, a geometric difference with first term u_1 and common ratio r has the general term u_n=u_1r^{n-1}, where r\neq 0,1,-1 and u_1\neq1.

  • The formula for the sum of a finite geometric progression is \displaystyle S_n=\frac{u_1(1-r^n)}{1-r},\quad r\neq1.

  • The derivation is as shown (note the cancellations):

    \begin{array}{ccccccccccccccc} \phantom{aaaaaaa}S_n&=&&&u_1&+&u_1r&+&u_1r^2&+&\ldots&+&u_1r^{n-1}\\ - \\ \phantom{aaaaaaa}rS_n&=&&&&& u_{1}r&+&u_{1}r^2&+&\ldots&+&u_1r^{n-1}&+&u_1r^n \\ \hline (1-r) S_n&=&&&u_1-u_1r^n& \end{array}

    \displaystyle \hspace{0.95cm} \implies \large S_n=\frac{u_1(1-r^n)}{1-r}, \quad (r\neq 1)

  • If \lvert r \rvert <1, then for large values of n, r^n approaches zero and the formula becomes \displaystyle S_n=\frac{u_1}{1-r}.

Proof

  • A proof in mathematics is an argument consisting of a logical set of steps that validates the truth of a general statement beyond any doubt.

Types of proofs
  • A direct proof is a method of proof that involves constructing a series of reasoned connected established facts.

  • To write a direct proof, you need to:

    • identify the given mathematical statement

    • use axioms, theorems, etc. to make deductions that prove the conclusion of a given statement.

\large \text{Example}

We can show that two numbers always sum up to an even number using direct proof.

Let m and n be two odd positive integers.

Then m=2p+1 and n=2q+1, where p,q \in \mathbb{Z}^+.

Therefore, m+n=2p+2q+2=2(p+q+1).

Since p+q+1 \in \mathbb{Z^+}, m+n is even.

  • A statement may not always be easily proved directly. In which case you need to employ a different type of proof.

  • Statements can also be proved using contradiction. For a proof by contradiction, the following steps are involved:

    • identify the implication of the given statement

    • assume that the implication is false

    • use axioms, theorems, etc. to produce a contradiction

    • conclude that the original statement is true.

\large \text{Example}

The number \sqrt{2} can be shown to be irrational using contradiction.

Assume (in search of a contradiction) that \sqrt{2} is rational that is \sqrt{2}=\frac{m}{n} where m,n \in \mathbb{Z}, m,n have no common factors and n\neq 0.

Then 2=\frac{m^2}{n^2}; so m^2=2n^2.

This means m^2 is an even integer since n^2 is an integer.

It follows that m is even. So m=2k and m^2=4k^2.

Therefore, m^2=4k^2=2n^2 which implies that n^2=2k^2. This means that n^2 is also an even integer.

Since m and n are even integers, they have a common factor of 2, which contradicts the assumption that m, n have no common factors.

  • An example which contradicts a given statement can be used to justify that a statement or conjecture is false.

  • Finding such an example is the basis of a proof known as counterclaim or counterexample.

\large \text{Example}

Consider the statement if n\in \mathbb{Z}, then n^2+1 is prime. For n=3,\ n^2+1=10 which is not a prime number. Thus, n=3 is a counterexample.

  • Yet another type of proof is proof by induction which makes use of the principle of induction.

  • Mathematical induction is a technique for proving that a statement about an integer $n$ is true for every integer n greater than or equal to some starting integer n_0.

  • Let P(n) be a statement pertaining each positive integer n. If

    • P(1) is true and

    • the implication: “If P(k) then P(k+1)” is true for every positive integer k,

      then P(n) is true for every positive integer n. This is the principle of mathematical induction.

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