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Geometric Series

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The numbers make foundation stone of the mathematics subject. We commonly come across with concepts like series and sequences that are wholly based upon the numbers. A sequence is a collection of objects, mostly numbers, in an ordered way. For example - A sequence of even numbers is 2, 4, 6, 8, ...
On the other hand, a series is said to be the sum of a sequence. We can say that if we place an addition sign (+) between every two numbers of a series, then it refers to a series.

For Instance: 
2 + 4 + 6 + 8 + ... becomes a series.

On the basis of sum of the terms, the series can be classified into two - convergent and divergent. If the limit of the partial sum of series is finite, then it is said to be convergent series. On the other hand, if the limit of sum does not exist or is infinite, then it is called divergent series.
There are several different types of series studied in mathematics. The kind of series that are first learnt in algebra are - arithmetic series, geometric series and harmonic series. We are going to focus on geometric series here. They are the most common examples of infinite series having finite sums (this condition is not necessarily satisfied for all geometric series).
The 
geometric series have been quite important in early calculus. They are most often seen in the study of convergence. These series are utilized not only in mathematics, but also in engineering, physics, economics, finance, queueing theory, computer science and even in biology. Let us go ahead and learn about geometric series in detail.

Definition

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A geometric series is a kind of series in which the ratio between any two successive terms is always constant throughout. In other words, a series is said to be geometric when any term divided its predecessor term, the ratio remains same. i.e. if $a_{n}$ be the n$^{th}$ term of the series, then

$\frac{a_{n+1}}{a_{n}}$ = constant

This constant is known as the common ratio and is denoted by letter "r".

For Example:

i)
2 + 4 + 8 + 16 + 32 + ...

Here, the common ratio is 2.

ii) 4 + 0.4 + 0.04 + 0.004 + 0.0004 + ยทยทยท

Here, common ratio is $\frac{1}{10}$.

Finite Geometric Series

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When the number of terms in a geometric series are finite, they are called finite geometric series. The general form of a finite geometric series is given below:

a + ar + ar$^{2}$ + ar$^{3}$ + ar$^{4}$ + ... + ar$^{n}$

Where, a is said to be the "first term" of the series and r be the "common ratio". Also, n is an integer.

We can also write it in the following way:

S = $\sum_{i=0}^{n}ar^{i}$

The finite geometric series possess first as well as the last term.

Infinite Geometric Series

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In the similar way, an infinite geometric series is defined as a geometric series having infinite number of terms in it . The general form of an infinite geometric series can be written as under:

a + ar + ar$^{2}$ + ar$^{3}$ + ar$^{4}$ + ...

Where, a , r and n have same definitions as above. This series can also be written in the compact as shown below:

S = $\sum_{i=0}^{\infty}ar^{i}$

The infinite geometric series continue indefinitely, they do not have any specified last term. These series can be commonly seen in mathematics and physics.

Formula

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The formulae that are being utilized in various calculations related to geometric series are illustrated in this section.

Formula for n$^{th}$ term:

When we need to find the value of a term at n$^{th}$ position (i.e. 4$^{th}$ term, 13$^{th}$ term, 100$^{th}$ term etc.), we use the following formula:
$T_{n} = a r^{n-1}$
Where, $T_{n}$ denotes n$^{th}$ term, a be the first term of the series, while r be the common ratio.

Formulae for Sum:

The formula for the sum of first n terms of a finite geometric series is as follows:
$S_{n} = \frac{a(1-r^{n})}{1-r}$
Here, r $\neq$ 1

The formula for the sum of first n terms of an infinite geometric series is:
$S_{\infty} = \frac{a}{1-r}$
provided that |r| < 1.

Convergence

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A series is said to be a convergent series if sequence made by its partial sums tend to a finite number.
 
Let us suppose that 

$S_n = \sum_{i=1}^{n} a_{i}$

be a series, then its partial sums would be given by {$S_{1}, S_{2}, S_{3}, ...$}.

If this sum converges or approaches to a certain number, them it is said to be convergent series.

An geometric series that is an infinite series, is said to be a convergent series if its common ratio is such that -1 < r < 1. If it does not satisfies this condition, it is called a divergent series.