# Finite and Infinite Series

A sequence containing finite number of terms is called a finite sequence and the series corresponding to this sequence is a finite series.

Example:
1, 3, 5, 7 is a finite sequence of four terms.
1+3+5+7 is finite series of 4 terms.
A sequence which is not a finite sequence is an infinite sequence and the series corresponding to an infinite sequence is an infinite series.

Example:
1, 3, 5, 7, ........ the set of all odd numbers is an infinite sequence.
1+3+5+7+......... is inifinite series.
Consider G.P: term becomes small, and approaches zero as gets very large.
We know that for a G.P with first term and common ratio .
When

Note:
1. If then sum of an infinite G.P approaches infinity.
2. is not possible for an A.P.
Example 1:
The sum of an infinite geometric series is 57. The sum of their cubes is 9747. Find the series.

Solution:

Example 2:
A ball dropped from a height of 100m, bounces to half the height with each bounce. Find the total distance covered by the ball till it comes to rest.

Solution:
(Recollect the example given in introduction?)

Example 3:
Evaluate

Solution:

Example 4:
Using infinite G.P, express as a fraction.

Solution:

Example 5:
In an infinite G.P, each term is equal to three times the sum of all the terms that follow it. The sum of the first two terms is 15. Find the sum of the series to infinity.

Solution:

Example 6:
Show that in an infinite G.P (in which ) each term bears a constant ratio to the sum of all the terms that follow it.

Solution: