How to Solve Sum of Infinite Series | Tougher Problem For IIT-JEE Advanced | Sequence and Series

Описание: In mathematics, a sequence is an ordered list of numbers, while a series is the sum of the terms in a sequence.

More formally, a sequence is a function whose domain is the set of natural numbers, and a series is the sum of the terms of a sequence, denoted by the symbol ∑ (sigma).

For example, the sequence {1, 2, 3, 4, 5, ...} is the sequence of natural numbers, and the series corresponding to this sequence is the sum of all the terms in the sequence, which is denoted by ∑n=1∞ n and is equal to infinity.

There are many types of sequences and series, including arithmetic sequences, geometric sequences, harmonic sequences, and Fibonacci sequences, among others. Each of these types of sequences and series has its own unique properties and formulas, which can be used to calculate the values of the terms in the sequence or the sum of the series.

Arithmetic sequences are sequences in which each term is obtained by adding a fixed constant, called the common difference, to the preceding term. For example, the sequence {1, 3, 5, 7, 9, ...} is an arithmetic sequence with a common difference of 2.

Geometric sequences are sequences in which each term is obtained by multiplying the preceding term by a fixed constant, called the common ratio. For example, the sequence {1, 2, 4, 8, 16, ...} is a geometric sequence with a common ratio of 2.

Harmonic sequences are sequences in which each term is the reciprocal of a term in an arithmetic sequence. For example, the sequence {1, 1/2, 1/3, 1/4, 1/5, ...} is a harmonic sequence.

Fibonacci sequences are sequences in which each term is the sum of the two preceding terms. For example, the Fibonacci sequence begins with {0, 1, 1, 2, 3, 5, 8, 13, 21, ...}.

Each type of sequence and series has its own formulas for calculating the sum of the terms in the sequence or the sum of a specific number of terms in the series. These formulas can be useful in a variety of mathematical and scientific applications, such as calculating interest on a loan, modeling population growth, and analyzing data.


The sum of an infinite geometric series with first term 'a' and common ratio 'r' is given by the formula:

S = a / (1 - r)

where S represents the sum of the infinite series.

This formula applies only if the absolute value of the common ratio 'r' is less than 1, which ensures that the series converges to a finite limit.

If the absolute value of the common ratio 'r' is greater than or equal to 1, the series diverges and does not have a finite sum.

To understand this formula, consider the series:

a, ar, ar^2, ar^3, ar^4, ...

To find the sum of the first n terms of the series, we can use the formula for the sum of a finite geometric series:

Sn = a(1 - r^n) / (1 - r)

Now, as n approaches infinity, the value of r^n approaches 0 if the absolute value of r is less than 1. Thus, the sum of the infinite series is given by:

S = lim ninf Sn = a / (1 - r)

So, if the absolute value of the common ratio 'r' is less than 1, the infinite series converges to the value of 'a' divided by 1 minus 'r'. If the absolute value of 'r' is greater than or equal to 1, the series diverges and does not have a finite sum.




Mathematics, Education, Learning, Math Lessons, Sequences, Series, Geometric Progression (GP), Infinite Series, Calculus, Algebra, Math Help