Infinite geometric series

An infinite geometric sequence is a special case of an infinite sequence, so let’s get to know an infinite sequence first.

An infinite series is the sum of an infinite number of terms of a sequence of numbers :

The general (last) term of infinite series is denoted by or .

An infinite series (which is also called sum) is denoted by the capital Greek letter, .

We write the sums of the first n terms of the sequence:

The obtained sums are called partial sums of the sequence. We denote them by . The sequence is called the sequence of subtotals.

The sum of the infinite sequence is the limit of the sequence of subtotals when . The limit of the sequence of partial sums is denoted by and we write:

Infinite geometric series

Suppose we have a given infinite geometric sequence and we want to calculate its sum . To calculate the sum of the given sequence, we use an infinite geometric series. Since this is geometric sequence, we know that: applies, so we can have:

written as:

We know that a series is convergent if there is a limit of partial sums, so we find the sum for the first n terms:

The notation is recognized as a finite geometric series, so it can be written shorter using the form for calculating the sum of the finite geometric series:

If the series is convergent, then there is a limit to its partial sums, denoted by .

Let's obtain a formular for :

We note that the limit of the partial sum will exist if the limit of the expression exist. So, consider when the limit of the expression exist.

The value of the power converge towards zero with increasing exponent n if the value of the ratio r is . Therefore:

For other values of r, the limit of the expression does not exist.

We conclude that the limit of partial sum exists if:

or

We calculate the limit of (1):