Geometric Series

Geometric sequence is

The geometric sequence is a sequence in which each term has a constant(common) ratio to its preceding term.
For example, the geometric sequence with the first term ‘a’ and the common ratio ‘r’ is as follows.

$$a,\quad ar,\quad a{ r }^{ 2 },\quad a{ r }^{ 3 },\quad a{ r }^{ 4 }…$$

If the common ratio is greater than ‘1’, the sequence value will increase.
For example, if the first term is ‘1’ and the common ratio is ‘2.’

$$1,\quad 2,\quad 2^{ 2 },\quad { 2 }^{ 3 },\quad { 2 }^{ 4 }…$$

If the common ratio is a value between 0 and 1, the sequence’s value becomes smaller and smaller.
For example, if the first term is ‘1’ and the common ratio is ‘0.5 (= 1/2)’,

$$1, 0.5, 0.5^{ 2 }, { 0.5 }^{ 3 }, 0.5^{ 4 }… = 1, \frac { 1 }{ 2 } ,\frac { 1 }{ 4 } , \frac { 1 }{ 8 } , \frac { 1 }{ 16 } …$$

Geometric sequences have the following characteristics according to the common ratio ‘r’.

1. r > 1 : The value of the term is increasingly larger.
2. r = 1 : All the terms are has same value.
3. 0 < r < 1 : The value of the term becomes smaller and becomes exponentially closer to 0.
4. r = 0 : All terms excluding the first term are zero.
5. -1 < r < 0 : The minus (-) sign is alternately displayed, and it approaches zero exponentially.
6. r = -1 : The absolute value of the term is the same, but the minus (-) sign is alternately displayed.
7. r < -1 : The value of the term becomes larger and the minus (-) symbol is displayed alternately.

Infinite geometric series

Continuing to add the terms of the geometric sequence, it becomes an infinite geometric series.
If the common ratio is 1 or more, the geometric series will infinitely increase. Conversely, if the common ratio’s absolute value is smaller than 1, the geometric series tends to approach a fixed value.