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}

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