# Sequences and Series

We use the words sequence and series interchangeably and we generally mean a continuation of numbers/events by them. In mathematics, however, these words have a particular meaning.

A sequence (also known as progression) is an arrangement of numbers in a specific order, in such a way that a definite relation exists between the numbers and their positions.

A series is what one gets, when all the terms of a sequence are added.

The numbers which make a sequence are known as the terms. The ${n}$th term is generally denoted by ${t_n}$.

Summing up the terms of the sequence ${t_1, t_2, t_3, t_4, t_5, \cdots , t_n}$, we get a series ${S_n}$.

It is easy to prove that ${t_n = S_n - S_{n-1}}$.

• ### WHY do we study sequences and series?

1) The investment on compound interest grows annually forming a geometric sequence.

2) Population models of bacteria (2 from 1, 4 from 2 etc.) follow a geometric sequence.

3) Difficulty level of problems increases in such a way that each problem takes 5 minutes more than the previous one to be solved. This follows an arithmetic sequence.

4) The harmonic sequence ${1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \frac 1 5 + \cdots}$ diverges to ${\infty}$, and is a fundamental result in mathematics. It has got applications in estimating traffic and in destroying noise!

Terms of a sequence form a function defined from ${\mathbb N}$, the set of natural numbers to ${t_n}$, the set containing the terms in the sequence; e.g. consider the sequence ${1,4,9,16,25, \cdots}$. Clearly, ${t_n = f(n) = n^2 \ \forall n \in \mathbb N}$.

• ### Arithmetic Progression (A.P.)

Arithmetic Progression is a sequence for which ${t_{n+1}-t_n}$ is constant ${\forall n \in \mathbb N}$. The constant difference , ${d}$, is known as the common difference and generally, the first term is denoted by ${a}$.

So, ${n}$th term will be ${a+ (n-1)}d$ and sum of first ${n}$ terms will be

${\frac {n}{2} [2a + (n-1)d]}$

• ### Properties of A.P.

Let ${k}$ be a non-zero number. Let ${a, a+d, a+2d, \cdots}$ be an A.P. Then,

I) ${a+k, a+d + k, a+2d + k, \cdots}$ is also an A.P. with first term ${a+k}$ and common difference ${d}$

II) ${ak, (a+d)k, (a+2d)k, \cdots}$ is also an A.P. with first term ${ak}$ and common difference ${dk}$.

• ### Geometric Progression (G.P.)

Geometric Progression is a sequence for which ${\frac {t_{n+1}}{t_n}}$ is constant ${\forall n \in \mathbb N}$. The constant ratio , ${r}$, is known as the common ratio.

If ${a}$ is the first term of a G.P., then the ${n}$th term is ${ar^{n-1}}$ and the sum of first ${n}$ terms is given by

${a \Big ( \frac {r^n-1}{r-1} \Big) , r \ne 1}$

• ### Properties of G.P.

Let ${k}$ be a non-zero number. Let ${a, ar, ar^2, \cdots}7s=1$ be a G.P. Then,

I) ${ka, kar, kar^2, kar^3, \cdots}$ is also a G.P. with first term ${ka}$ and common ratio ${r}$

II) ${a^k, (ar)^k, (ar^2)^k, (ar^3)^k, \cdots}$ is also a G.P. with first term ${a^k}$ and common ratio ${r^k}$.

Having studied limits, we can show that if ${|r| < 1}$, then sum of all terms of a G.P. (up to infinity) is given by

${\frac {a}{1-r}}$

In other words,

${\lim \limits_{n \to \infty} S_n = \frac {a}{1-r}}$

• ### Harmonic Progression (H.P.)

Harmonic Progression is a sequence such that reciprocals of its terms are in arithmetic progression. Thus, if ${t_1, t_2,t_3, \cdots, t_n, \cdots}$ is a harmonic progression, then

${\frac {1}{t_1}, \frac {1}{t_2}, \frac {1}{t_3}, \cdots, \frac {1}{t_n} , \cdots}$

is an A.P.

Arithmetic mean, ${A}$, of ${x}$ and ${y}$ is given by ${\frac {x+y}{2}}$. It can be shown that

${x, \frac {x+y}2, y}$

are in A.P.

Geometric mean,${G}$, of ${x}$ and ${y}$ is given by ${\sqrt {xy}}$. It can be shown that

${x, \sqrt {xy}, y}$

are in G.P.

The harmonic mean, ${H}$, of ${x}$ and ${y}$ is given by ${\frac {2xy}{x+y}}$. Thus,

${\frac 1 x , \frac {x+y}{2xy}, \frac 1 y}$

are in A.P.

It can be shown that

${G^2 = AH \ and \ A \ge G \ge H}$

An arithmetico-geometric progression is a combination of an A.P. and a G.P. It is of the form

${a, (a+d)r, (a+2d)r^2, (a+3d)r^3 , (a+4d)r^4}$

Sum of first ${n}$ terms of such a progression is given by

${S_n = \frac {a}{1-r} + \frac {dr(1-r^{n-1})}{(1-r)^2} - \frac {r^n [a+(n-1)d]}{1-r}}$

• ### The summation notation ${\sum}$

Let there be a sequence ${t_1, t_2, t_3, t_4, \cdots, ... t_m}$. The sum of these $m$ terms is briefly written as

${\sum \limits_{r=1}^m t_r = t_1 + t_2 + t_3 + t_4 + \cdots + t_m}$

• Properties of the summation notation

${\sum \limits_{r=1}^m at_r = a \sum \limits_{r=1}^m t_r}$

${\sum \limits_{r=1}^m a_r + b_r \sum \limits_{r=1}^m a_r + \sum \limits_{r=1}^m b_r}$

• Important results :

I) Sum of first ${n}$ natural numbers is given by

${\frac {n(n+1)}{2}}$

II) Sum of squares of first ${n}$ natural numbers is given by

${\frac {n(n+1)(2n+1)}{6}}$

III) Sum of cubes of first ${n}$ natural numbers is given by

${\frac {n^2(n+1)^2}{4}}$

• ### Exponential and logarithmic series

I) The exponential series ${e^x}$ is an infinite series given by

${\sum \limits_{r=0}^{\infty} \frac {x^r}{r!} = 1 + \frac {x}{1!} + \frac {x^2}{2!} + \frac {x^3}{3!} + \cdots + \frac {x^n}{n!} + \cdots }$

II) For ${|x| < 1}$,

${ln (1+ x) = \sum \limits_{r=1}^{\infty} (-1)^{r+1} \frac {x^r}{r} = x - \frac {x^2}{2} + \frac {x^3}{3} - \frac {x^4}{4} + \cdots}$