# Infinite Series Formula

## Infinite Series Formula

The sum of a sequence with infinitely many terms is determined using the Infinite Series Formula. Infinite series come in a variety of forms. The sum of infinite geometric series and the sum of infinite arithmetic series will be covered in this section. The geometric series is the sequence where the ratio of the subsequent terms to the preceding term is the same throughout, whereas the arithmetic series is the sequence where the difference between each consecutive term is constant throughout. The Infinite Series Formula can be used to quickly calculate the sum.

## What Is Infinite Series Formula?

To calculate the sum of a series that goes on forever, one uses the sum of the infinite geometric series formula. It is also referred to as the total of infinite GP. Although the series has infinite terms, one discovers that the sum of a GP converges to a value. If -1<r<1, the Infinite Series Formula can be written as,

Sum = a / (r – 1) Where,

A is the series’ initial term.

R stands for the common ratio of two successive words, and -1< r <1.

Given that the terms are in an infinite geometric progression with a common ratio absolute value less than 1, the infinite series formula is used to calculate the total of an infinite number of terms. This is due to the fact that the total will only converge to a specific value if the common ratio is smaller than 1, else the absolute value of the sum will tend to infinity.

Divergent series describes an infinite series with the natural numbers 1 through 4 as its terms. The triangular number is the series’ nth partial sum.

Although the series initially appears to have no significance, it can be used to produce a number of mathematically intriguing results. Mathematicians, for instance, use a variety of summation techniques to give numerical values to even divergent series. In particular, the Ramanujan summation and zeta function regularisation procedures give the series a value of – 1 / 12, which is denoted by a well-known formula.

1+2+3+4+…… = -1/12.

where the left-hand side must be regarded as the result of one of the aforementioned summation techniques rather than as the total of an infinite series in the conventional sense. Other disciplines, such as complex analysis, quantum field theory, and string theory, can use these techniques.

University of Alberta mathematician Terry Gannon describes this equation as “one of the most astonishing formulae in science” in a monograph on moonshine theory.

The sum of an infinite number of numbers, variables, or functions that adhere to a particular rule is known as an infinite series. Infinite series are minor characters in the calculus drama. Infinite series respectfully stand to the side as derivatives and integrals rightfully take centre stage. Near the end of the course, when everyone is struggling to reach the finish line, they finally appear. Infinite series are useful for demonstrating subtle aspects of mathematical rigour as well as for approximating solutions to challenging situations.

### Examples using Infinite Series Formula

Infinite Series Formula is important from the exam perspective. It is crucial to practice questions related to the Infinite Series Formula. All the questions based on the Infinite Series Formula can be solved easily by referring to the NCERT solutions provided by Extramarks. Students must practice exercises of each chapter in Mathematics in order to perform well in their upcoming examinations. The study materials available on Extramarks are of high quality and very reliable. All the questions given in the Mathematics textbook are important because they can appear in the final examination. Practising them on a regular basis is a must. Extramarks has all the required study materials that are helpful in scoring higher marks in the examinations. Students need to keep referring to them during their preparations. It is advisable to keep referring to the syllabus because questions are framed from the syllabus itself.