Series Formula

In mathematics, a series has a constant difference between terms. We can find out the sum of the terms in arithmetic series by multiplying the number of times the average of the last and first terms. Thus we can see that series and finding the sum of the terms of series is a very important task in mathematics. In this topic, we will discuss some popular series and series formula with examples. Let us learn the concept!

What is the Series?

What exactly is a series? Actually, a series in math is simply the sum of the various numbers or elements of the sequence. For example, to make a series from the sequence of the first five positive integers 1, 2, 3, 4, 5 we will simply add them up. Therefore 1 + 2 + 3 + 4 + 5 is a series.

So, series of a sequence is the sum of the sequence to some given number of terms, or sometimes till infinity. It is often written as S_n.

If the sequence is 2, 4, 6, 8, 10, … , then the sum of the first 3 terms:

\(S_3  = 2 + 4 + 6\)

\(S_3 = 12.\)

The Greek capital sigma i.e. \(\Sigma\) is usually used to represent the sum of a sequence. This may be best explained using the example: \(\sum_{i=1}^{4}3i\)

This means to replace the term ‘i’ in the expression by 1 and write down what we get. Then replace ‘i’ by 2 and write down what we get. Keep doing this until we get to 4 since this is the number above the Sum.

Source:en.wikipedia.org

Some Popular Series

1] Arithmetic Progressions

An arithmetic progression is a sequence, in which each term is a certain number larger than the previous term.  The terms in the sequence are known as to increase by some common difference, d.

In general, the nth term of the arithmetic progression, given the first term ‘a’ and common difference ‘d’ will be \(a_n = a + (n – 1)d\)

2]Geometric Progressions

A geometric progression is a sequence in which each term is r times larger than the previous term. Here r is called the common ratio of the sequence. Then the nth term of a geometric progression is as given : \(a_n = ar^{n-1}\)

The first term is denoted as ‘a’  and the common ratio is denoted as ‘r’

3]Harmonic Progressions

Series as \(\frac{1}{1} , \frac{1}{2} , \frac {1}{4}\), ….. is called harmonic progression. It is having terms inverse of its corresponding arithmetic progression.

Its nth term with first term ‘a’ and common difference‘d’ will be, \({a_n} = \frac{1}{{a + (n – 1)d}}\)

Series Formula

1] The sum to n terms of an arithmetic progression

This is given by: \({S_n} = \frac{n}{2}[2a + (n – 1)d]\)

Also, \({S_n} = \frac{n}{2}(a + l)\)

Where l is last term.

2] The sum of a geometric progression

The sum of the n terms of a geometric progression is given as:

\({S_n} = \frac{{a(1 – {r^n})}}{{1 – r}};{\text{ }}r \ne 1\)

\({S_n} = \frac{{a – rl}}{{1 – r}};{\text{ }}l = a{r^n}\)  where r < 1

3] The sum of a geometric progression with infinite terms

The sum S of infinite geometric series will be

\(S = \frac{a}{{1 – r}};{\text{ }}\left| r \right| < 1\)

Solved Examples for Series Formula

Q: Find the sum of the following series:

\(\frac{1}{2} , \frac{1}{4} , \frac{1}{8}\) ,……

Solution: It is a geometric progression with infinite terms.

Here \(a = \frac{1}{2}\)

And \(r = \frac{1}{2} < 1\)

Sum, \(S = \frac{a}{{1 – r}}\)

i.e. \(S = \frac{\frac{1}{2}}{{1-\frac{1}{2}}}\)

i.e. S = 1

Therefore, the sum to infinity terms = 1.