Special Series

What is an arithmetic series and how do you find its sum?

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. To find the sum of the first nnn terms of an arithmetic series, you can use the formula: S_n = n²(2a + (n – 1)d)

How do you calculate the sum of a geometric series?

A geometric series is a sequence of numbers where each term is multiplied by a constant ratio to get the next term. The sum of the first nnn terms of a geometric series can be calculated using the formula:

[Tex]S_n = a \frac{r^n – 1}{r – 1} [/Tex]

What is a harmonic series and where is it used?

A harmonic series is a sequence of numbers where each term is the reciprocal of an integer. The nnnth harmonic number is the sum of the reciprocals of the first nnn natural numbers:

Hⁿ = 1 + 1/2 + 1/3 + 1/4 ………..

What is the Fibonacci series and how is it generated?

The Fibonacci series is a sequence of numbers where each term is the sum of the two preceding ones, starting from 0 and 1. The first few terms of the Fibonacci series are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.

Special Series: A series can be defined as the sum of all the numbers of the given sequence. The sequences are finite as well as infinite. In the same way, the series can also be finite or infinite. For example, consider a sequence as 1, 3, 5, 7, … Then the series of these terms will be 1 + 3 + 5 + 7 + ….. The series special in some way or the other is called a special series. The following are the three types of special series.

1. 1 + 2 + 3 +… + n (sum of first n natural numbers)
2. 1² + 2² + 3² +… + n² (sum of squares of the first n natural numbers)
3. 1³ + 2³ + 3³ +… + n³ (sum of cubes of the first n natural numbers)

In this article, we will read about special series in maths and how to get the formula for all these series. We will also see solved examples and practice problems on special series.