Notes On Sequences and Series - CBSE Class 11 Maths
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 + a3 + 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 + a3 + a4 + ... + an + ... can be abbreviated as ∑∞n=1an . Similarly, in summation notation, the finite series a1 + a2 + a3 + a4 + ... + an + ... can be written as ∑∞k=1ak ∑∞n=1an or ∑∞k=1ak are known as the compact forms of a series.

#### 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 + a3 + 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 + a3 + a4 + ... + an + ... can be abbreviated as ∑∞n=1an . Similarly, in summation notation, the finite series a1 + a2 + a3 + 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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