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.