Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty} 5(0.45)^{n}$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is approximately 9.09.

Step 1: Identify the first term and the common ratio

The first term (a) of the series is 5, and the common ratio (r) is 0.45.

Step 2: Check if series is summable

Verify whether the absolute value of the common ratio is less than 1 (|r| < 1). In this case, |0.45| is indeed less than 1, which indicates that the sum of the series does exist.

Step 3: Calculate the sum

Use the sum formula \(S = \frac{a}{1 - r}\) to calculate the sum. Substituting the values a = 5 and r = 0.45 gives us \(S = \frac{5}{1 - 0.45} = 9.09090909091\).

Key Concepts

Sum of Series

In the world of geometric series, when we talk about the "sum of the series," we are referring to adding up all the terms of a series to reach a particular total. An infinite geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio.
The magic of figuring out the sum of an infinite geometric series lies in whether it's possible to calculate a finite sum. If the absolute value of the common ratio is less than 1, the series will converge, meaning you can sum its infinite terms to get a single number. However, if the absolute value of the common ratio is greater than or equal to 1, the series will not converge, leading to an undefined sum.
For the series in our example, the first term is given as 5, and the calculation uses the formula \(S = \frac{a}{1 - r}\) to find the sum, where \(a\) is the first term. By substituting \(a = 5\) and \(r = 0.45\), the sum turns out to be approximately 9.09. This means that even though there are infinitely many terms, they sum up to about 9.09, showcasing the neat efficiency of mathematics. Always remember, for a sum to exist in infinite geometric series, \(|r|\) must be less than 1.

Common Ratio

The "common ratio" is a fundamental characteristic of any geometric series and plays a big role in determining whether the series can be summed. Put simply, the common ratio \(r\) is the factor you multiply each term by to get the next one in the series.
In our series example, the common ratio is \(r = 0.45\). The reason this particular series can be summed is that the common ratio's absolute value is less than 1 (i.e., \(|0.45| < 1\)). This condition ensures the terms in the series get progressively smaller, allowing the total sum to approach a fixed number as more terms are added.
When dealing with any geometric series, identifying the common ratio is one of the first steps. It always helps to check whether the series converges by verifying if \(|r| < 1\). If it does, you're on the right track to finding a predictable and precise sum. Remember, the smaller the common ratio, the faster each term decreases, and the more it contributes to a manageable sum.

Geometric Series Formula

The "geometric series formula" is a handy tool that simplifies the process of summing the terms of an infinite geometric series. The specific formula used for summing an infinite series that converges is: \(S = \frac{a}{1 - r}\).
In this formula, \(a\) is the first term of the series, and \(r\) is the common ratio. This neat formula can only be applied if \(|r| < 1\), allowing for the infinite series to have a defined sum.
To use the geometric series formula effectively, always start by confirming that the series is summable (i.e., it converges). In the example provided, with a first term \(a = 5\) and common ratio \(r = 0.45\), applying the formula gives: \[S = \frac{5}{1 - 0.45} = 9.09090909091\].
This formula simplifies what could be a tedious process of adding up an infinite number of terms into a straightforward calculation. The beauty of the formula lies in its simplicity and effectiveness, allowing learners to tackle complex series with ease. Always ensure you confirm the condition \(|r| < 1\) before applying the formula to ensure valid results.