Calculates Sum of Infinite Geometric Series. When the first and ratio, r terms are given.\[S_{\propto}=\frac{a}{1 - r}\]
Let's say we have a geometric series with a first term, a, and a common ratio, r. The formula for finding the sum of this infinite geometric series is:
S = a / (1 - r),
where S represents the sum of the series.
To understand why this formula works, let's manipulate it a bit. Multiply both sides of the equation by (1-r):
S(1 - r) = a.
Expanding the left side, we get:
S - rS = a.
Rearranging the terms:
S = a / (1 - r).
This formula allows us to find the sum of an infinite geometric series.
However, there is a condition that must be met for this formula to work - the absolute value of the common ratio, |r|, must be less than 1.
If the common ratio has an absolute value less than 1, it means that each subsequent term in the series gets progressively smaller, tending towards zero. As a result, the sum of the series will converge to a finite value.
To understand this, imagine a geometric series where the common ratio is, for example, 0.5.
The first term is a, the second term is a 0.5, the third term is a 0.5^2, and so on.
As the exponent increases, the value of 0.5^2 becomes smaller and smaller (0.5^2 < 0.5^1 < 0.5^0). If we continue this pattern indefinitely, the terms will approach zero.
The sum of an infinite geometric series with a common ratio less than 1 can be thought of as the limit of the series as the number of terms approaches infinity. This limit will yield a finite value.
