How to Find the Sum of an Infinite Geometric Series

Understanding the concept of an infinite geometric series is essential in various fields of mathematics and science. This article delves into the process of finding the sum of such a series using a specific example: the sum of the series ( S frac{3}{4} frac{3}{4}^2 frac{3}{4}^3 ldots infty ).

Understanding Infinite Geometric Series

An infinite geometric series is a sequence of numbers where each term is a constant multiple of the previous term, and the sum of the series can be calculated under certain conditions. The general form of an infinite geometric series is: [ a ar ar^2 ar^3 ldots ] where a is the first term and r is the common ratio.

The Process of Finding the Sum

To find the sum of the given series, let's start with the expression for the series:

[ S frac{3}{4} frac{3}{4}^2 frac{3}{4}^3 ldots infty ]

We can simplify this by noting that each term after the first is a multiple of the previous term by the ratio ( frac{3}{4} ). Let's multiply both sides of the equation by the common ratio ( frac{3}{4} ):

[ frac{3S}{4} frac{3}{4}^2 frac{3}{4}^3 frac{3}{4}^4 ldots infty ]

Now, we can subtract the second equation from the first to eliminate the repeating terms:

[ S - frac{3S}{4} frac{3}{4} left( frac{3}{4}^2 - frac{3}{4}^2 right) left( frac{3}{4}^3 - frac{3}{4}^3 right) ldots ]

This simplifies to:

[ frac{S}{4} frac{3}{4} ]

Multiplying both sides by 4, we get:

[ S 3 ]

General Formula for the Sum

The sum of an infinite geometric series can be calculated using the formula:

[ S frac{a}{1 - r} ]

where a is the first term of the series and r is the common ratio. In our example, the first term ( a frac{3}{4} ) and the common ratio ( r frac{3}{4} ). Plugging these values into the formula, we get:

[ S frac{frac{3}{4}}{1 - frac{3}{4}} frac{frac{3}{4}}{frac{1}{4}} 3 ]

Example Series

The series we are considering is 0.75, 0.5625, 0.421875, 0.31640625, 0.2373046875, 0.17797851562, 0.13348388671875, 0.10011291478955078, 0.07508468609216907, 0.05631351471912632, … to infinity. The sum of this series can be calculated as follows:

[ S frac{frac{3}{4}}{1 - frac{3}{4}} frac{frac{3}{4}}{frac{1}{4}} 3 ]

Conclusion

By understanding the principles of infinite geometric series, we can easily calculate the sum using the given formula. The sum of the series ( S frac{3}{4} frac{3}{4}^2 frac{3}{4}^3 ldots infty ) is indeed 3. This method can be applied to any infinite geometric series where the common ratio is between -1 and 1.

Additional Resources

For further reading and exploration, you can refer to textbooks on advanced algebra or online resources that provide detailed explanations and problem sets on geometric series. Understanding these concepts is crucial in fields such as calculus and physics, where the sum of series is used to model real-world phenomena.