Geometric Series

Geometric Series:

In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series

is geometric, because each term except the first can be obtained by multiplying the previous term by \frac{1}{2} \ .

Geometric series are one of the simplest examples of infinite series with finite sums. Let us understand the role of geometric series,historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.
A sequence of non negative numbers is called a geometric sequence where the ratio of of each term to its preceding term is the same except for the first term. A geometric series is a series whose terms are in geometric sequence.
There is an interesting story related with geometric series. It is said that the King of Persia was very impressed with the inventor of the board game chess ( believed to be from India ) that he offered to give him any reward. The inventor wanted one grain of wheat to be placed on the first square of the chessboard , two grains on the second square , four on the third , eight on the fourth and so on. This demand of the inventor seemed very small to the King.
We can see that the grains placed in the various squares are the terms of a geometric sequence 1, 2 ¹, 2², 2³ , 2⁴, ... 2 ⁶³
Sum of grains = S ₆₄ = 1 + 2 ¹ + 2 ² + 2 ³ + ...... + 2 ⁶³
We take the reverse order S ₆₄ = 2 ⁶³+ 2 ⁶² +.....+ 2 ³ + 2 ² + 2 + 1 ------------> 1
2 * S ₆₄ = 2 ⁶⁴ + 2 ⁶³ +.....2 ⁴ + 2 ³ + 2 ² +2 + 1 ----------->2
Subtract 1 from 2 S ₆₄ = 2 ⁶⁴ - 1
A huge amount of grain.