Skip to main content

Sequence and Series

Sequence and Series

We have heard about sequence and Series. But before diving deep into Sequence and Series, let us understand what is a sequence and what is a series?

There are cases where even a small discrepancy in measurements can lead to a big trouble. In all other cases, measuring something is made easier by approximations. And as mathematicians, we like to do that a lot! To measure a quantity x, let us write several approximations a1 , a2 , a3…

where a2 is a better approximation of x than a1,a3 is better approximation of x than a2 and so on. This list of approximations that we just created is example of a sequence.

Definition 1: A sequence is the set of the outputs of a function defined from the set of natural numbers to the set of real numbers or complex numbers. If the co-domain of the function is the set of real numbers, it is called a real sequence and if it is the set of complex numbers on the other hand, it is called a complex sequence.

In very simple terms, a sequence is an ordered set of numbers. A sequence is denoted using braces. For e.g. the sequence of the approximations that we created can be denoted by {an}. The sequence {n}∞n = 1 represents the all natural numbers. A sequence can be finite or infinite depending upon the number of terms it can have.

If all the terms of a sequence eventually approach the value x, we say that the sequence converges to the limitx.We call that sequence convergent and we represent it as

limn→∞{an} = x

On the other hand, if a sequence doesn’t converge, it is divergent. Some examples of convergent and divergent sequences are:

{1n}∞n = 1 is a convergent sequence because limn→∞{1n} = 0
0,1,0,1,0,1…is a divergent sequence because it can’t converge to either 0 or 1.
The Fibonacci Sequence 1,1,2,3,5,8,13,… is a divergent sequence.
Let us now go ahead and understand series. Sequence and series are very often confused with each other. Series are derived from sequences.

Definition 2: A series is defined as the sum of the terms of a sequence. It is denoted by

∑i = 1∞ai

where ai  is the ith term of the sequence

i is a variable

∑ is a symbol which stands for ‘summation’. It was invented by Leonard Euler, a Swiss mathematician.

The above written expression basically means

 Sequence and series

For a series defined above, sequence of partial sums is defined as

a1 , a1 + a2 , a1 + a2 + a3 , a1 + a2 + a3 + a4,…

A series can be finite or infinite depending upon the number of terms it has. If the sequence of the partial sums of a series is a convergent sequence, we say that the series is convergent. This is represented as:

∑i = 1∞ ai = limn→∞ ∑i = 1n ai = x ,where x ∈ R

Otherwise, the series is said to be divergent. Some examples of series are:

1 + 2 + 3 + 4 + 5 +⋯ which is a divergent series.
12 + 14 + 16 + 18 +⋯ which is a convergent series.
Progression:

It is just another type of sequence. The only difference between a progression and a sequence is that there is a general formula which can be derived for representing the terms of a progression whereas it can’t be always done for a sequence. Sequences may be based on logical rule. Some examples which support this argument are:

2,3,5,7,9,…is the sequence of prime numbers. A formula for its general term can’t be derived.
2,4,6,8,10,…is a sequence of multiples of 2 which is also a progression since the general term (nthterm)can be represented by 2n.
3,9,27,81,…is a sequence of powers of 3 which is also a progression since the general term (rth term) can be represented by 3r..
Note: All progressions are sequences but converse may or may not be true.

There are several kinds of progressions viz.arithmetic progression, geometric progression, harmonic progression, etc. 
Labels:

Comments

Post a Comment

Popular posts from this blog

SSC Practice Problem Solution

SSC Practice Problem Solution In this post, we will discuss solution of  SSC Practice Problem . Question 1  The efficiency of (A + B) = 100/24 = (25/6)% The efficiency of (A + B + C) = 100/8 = (25/2)% The efficiency of C = 25/2 – 25/6 = (50/6)% Hence, C can alone finish this job in = 100/(50/6) = 12 days Answer: 12 days option (1) Question 2  Original price = Rs. 6000; Price after discount = 6000 – 1200 = Rs. 4800; Price after raising service contract = 4800 + 480 = Rs. 5280 Answer: Rs. 5280 option (2) Question 3  Suppose the received money by A, B, and C is respectively 5x, 6x, and 9x. 5x = 450; => x = 90; Hence, the total money = 20x = 20*90 = Rs. 1800; Answer: Rs. 1800 option (2) Question 4 Suppose the Cost price of the bag= Rs. x; Hence, x + 0.15x = 230; => x = Rs. 200; Selling price after selling on 20% = 200 + 20% of 200 = Rs. 240 Answer: Rs. 240 option (3) Question 5  The required percentage ...
Post a Comment
Read more

Average

Average In this post, We will learn about average.  Average of any collection of data is the sum of data divide by number of data. Average = (Sum of observations /  Number of observations) Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.  problems The arithmetic mean of x, y and z is 80, and that of x, y, z, u and v is 75, where u=(x+y)/2 and v=(y+z)/2. If x ≥ z, then the minimum possible value of x is (CAT 2018) As specified in the question mean of x and y and z is 80 x+y+z=3*80 mean of x, y, z,u,v is 75 x+y+z+u+v=75*5 2u=x+y 2v=y+z x+y+z+((x+y+y+z)/2)=75*5 y+3(x+y+z)=75*10 y=750-720 y=30 since x+y+z=240 y=30 so x+z=210 since x>=z so minimum value of x=210/2 = 105 Problem2: In first 5 inings sehwag has scored at average of 40. In sixth ining He had scored 52 what is His average after 6th ining. one simple approach is getting total score divided by no of inin...
Post a Comment
Read more

Complex Number

Complex Number While solving the equation of the form  a x 2 + b x + c = 0 , the roots of the equations can take three forms which are as follows: Two Distinct Real Roots Similar Root No Real roots (Complex Roots) The introduction of complex numbers in the 16th century made it possible to solve the equation  x 2 + 1 = 0 . The roots of the equation are of the form  x = ± − 1 ‾ ‾ ‾ √  and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots. We denote  − 1 ‾ ‾ ‾ √  with the symbol i, where i denotes Iota (Imaginary number). An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = b. Algebraic Operation on Complex numbers: Addition of two complex numbers Subtraction of two complex number Multiplication of two complex number Division of two complex number Power of Iota (i) ...
Post a Comment
Read more

Logarithm

Logarithm When a x   = N , then we say that x = logarithm of N to the base a and write it as x = log a N . In simple words, it represents the power to which a number must be raised. Let me expand on that by giving a simpler example. If we are asked, what would be the result if ‘a’ is multiplied with itself ‘b’ times; then your answer would be x = a*a*a*…. b (times). This can also be written as a^b. This is also known as ‘a raised to the power of b’ If we are asked, which number multiplied with itself ‘b’ times, will result in a; then you are asked for the value of x such that x*x*x*… b(times) = a => x^b = a => x = a^(1/b) This is also known as ‘bth root of a’ If swe are asked, how many times should you multiply ‘a’ with itself to get ‘b’, that is where the concept of logarithm comes into the picture. You are asked for the value of ‘x’ such that a*a*a…. x (times) = b => x = Log  a  b This is also known as ‘ Log b to the base a’ ...
Post a Comment
Read more