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.