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 th term is generally denoted by .
Summing up the terms of the sequence , we get a series .
It is easy to prove that .
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 diverges to , 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 , the set of natural numbers to , the set containing the terms in the sequence; e.g. consider the sequence . Clearly, .
Arithmetic Progression (A.P.)
Arithmetic Progression is a sequence for which is constant . The constant difference , , is known as the common difference and generally, the first term is denoted by .
So, th term will be and sum of first terms will be
Properties of A.P.
Let be a non-zero number. Let be an A.P. Then,
I) is also an A.P. with first term and common difference
II) is also an A.P. with first term and common difference .
Geometric Progression (G.P.)
Geometric Progression is a sequence for which is constant . The constant ratio , , is known as the common ratio.
If is the first term of a G.P., then the th term is and the sum of first terms is given by
Properties of G.P.
Let be a non-zero number. Let be a G.P. Then,
I) is also a G.P. with first term and common ratio
II) is also a G.P. with first term and common ratio .
Having studied limits, we can show that if , then sum of all terms of a G.P. (up to infinity) is given by
In other words,
Harmonic Progression (H.P.)
Harmonic Progression is a sequence such that reciprocals of its terms are in arithmetic progression. Thus, if is a harmonic progression, then
is an A.P.
Arithmetic mean, , of and is given by . It can be shown that
are in A.P.
Geometric mean,, of and is given by . It can be shown that
are in G.P.
The harmonic mean, , of and is given by . Thus,
are in A.P.
It can be shown that
An arithmetico-geometric progression is a combination of an A.P. and a G.P. It is of the form
Sum of first terms of such a progression is given by
The summation notation
Let there be a sequence . The sum of these $m$ terms is briefly written as
- Properties of the summation notation
- Important results :
I) Sum of first natural numbers is given by
II) Sum of squares of first natural numbers is given by
III) Sum of cubes of first natural numbers is given by
Exponential and logarithmic series
I) The exponential series is an infinite series given by
II) For ,