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Sequences and Series

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Sequences and Series - Lesson Summary

When a collection of objects is listed in a sequential manner or a definite order such that every member has a definite position, that is, it comes either before or after, every other member, it is called a sequence.

A collection of numbers arranged in a definite order according to some definite rule is called a sequence.

The members or numbers that are listed in the sequence are called its terms.

The terms of a sequence are denoted by a1, a2, a3,…,an,…

The nth term of a sequence is denoted by an. an is also known as the general term of the sequence.

Finite or infinite sequence:

If the number of terms in a sequence is finite or countable, then it is called a finite sequence.

If the sequence goes on forever or has an uncountable number of terms, then it is called an infinite sequence.

Ex: 1, 2, 3, 4, ... is an infinite sequence. 2, 4, 6, 8, 10 is a finite sequence.

Terms of a sequence can be expressed by an algebraic formula.

Consider the sequence 1/2, 1/4, 1/8, 1/16,....

The first term can be written as a1 = 1/2 = 1/21;

The second term can be written as a2 = 1/4 = 1/22;

The third term can be written as a3 = 1/8 = 1/23 ;

The fourth term can be written as a4 = 1/16 = 1/24, and so on.

nth term of this sequence can be expressed as an = 1/2n, where n is a natural number.

Any term of the sequence can be found by substituting the term value in place of n.

e.g. a8 = 1/28 = 1/256

In some cases, an arrangement of numbers in a sequence like {1,1, 3, 7, 17, 41, ………} has no apparently visible pattern, but the terms of this sequence are generated by the relation given by
a1 = a2 =1

a3 = a1 + 2a2

a4 = a2 + 2a3, and so on.

The nth term of this sequence is an = an-2 + 2an-1, n>2.

Sequence {1, 1, 2, 3, 5, 8 …} can be described as:

a1 = a2 =1;

a3 = a1 + a2;

a4 = a2 + a3 and an = an-2 + an-1, n >2.

This sequence is known as the Fibonacci sequence.

Sometimes, it is practically impossible to express the terms of a sequence by means of a particular algebraic formula. Such sequences can only be described verbally, like the sequence of prime numbers {2,3,5,7,………..}.

A sequence is a collection of all images of a function, x, from N or any subset of N to X, where N is the set of all natural numbers and X is any set, may be of objects, events or numbers.

Domain of function x = N

{ x(1), x(2), x(3),  x(4), …} Or {x1,x2,x3,x4,...}, where  x(1)= x1, x(2)= x2, x(3)= x3, x(4) = x4 ,

Sequences are also called progressions because the terms in a sequence progress in a definite manner obeying some specific algebraic rule.

Series

Let {a1, a2, a3, a4,...} be a sequence.

Then the expression a1 + a2 + a+ a4 + ... + an + ... is called the series corresponding to the given sequence.

The expression obtained on adding the terms of a sequence is called a series.

Here, a1 is called the first term, a2 the second term… and an the nth term of the series.

A series is finite if the given sequence is finite, and infinite if the given sequence is infinite.

In summation notation, the infinite series a1 + a2 + a+ a4 + ... + an + ... can be abbreviated as ∑n=1an .

Similarly, in summation notation, the finite series a1 + a2 + a+ a4 + ... + an + ... can be written as ∑k=1ak

n=1an or ∑k=1ak are known as the compact forms of a series.

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