Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Find the sum of the infinite geometric series if it exists. $$1.5+0.015+0.00015+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is approximately 1.515.

Step 1: Identify the Series Terms

First, identify the first term and the common ratio of the geometric series. The series is given as \(1.5 + 0.015 + 0.00015 + \cdots\). The first term \(a\) is 1.5, and the second term is 0.015.

Step 2: Calculate the Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \(r = \frac{0.015}{1.5} = 0.01\). So, the common ratio \(r\) is 0.01.

Step 3: Evaluate Convergence

A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. Here, \(|r| = 0.01 < 1\), so the series converges.

Step 4: Apply the Infinite Geometric Series Formula

For a converging series, the sum \(S\) can be calculated using the formula \(S = \frac{a}{1 - r}\). Substitute \(a = 1.5\) and \(r = 0.01\) into the formula: \[S = \frac{1.5}{1 - 0.01} = \frac{1.5}{0.99}\].

Step 5: Calculate the Sum

Simplify the expression to find the sum: \[S = \frac{1.5}{0.99} = 1.51515\ldots\]. This is a repeating decimal, which can be rounded to 1.515 or expressed as \(\frac{151.5}{99}\) in fraction form.

Key Concepts

Geometric Series Convergence

One fascinating aspect of geometric series is how we determine their convergence. Convergence refers to whether the series will tend to a particular value as more terms are added. In the context of infinite geometric series, this occurs when the series approaches a finite sum.
To discern if a geometric series converges, we must examine the common ratio \(r\). The rule of thumb here is straightforward: a geometric series converges if the absolute value of \(r\) is less than 1: \(|r| < 1\).
This condition is critical because when \(|r|\) is less than 1, the terms in the series become progressively smaller, allowing the series to settle around a finite sum. If \(|r|\) is 1 or greater, the series either doesn't settle or blows up to infinity.
In our example, the common ratio \(r\) is 0.01, which meets the convergence condition \( |0.01| < 1 \). Thus, our series converges.

Sum of Infinite Series

Calculating the sum of an infinite geometric series might sound intimidating, but it’s quite manageable with the right formula.
For a converging geometric series, the sum \(S\) can be calculated using:

• \(S = \frac{a}{1 - r}\)

where \(a\) is the first term of the series, and \(r\) is the common ratio.
In our case, the first term \(a\) is 1.5, and the common ratio \(r\) is 0.01, thus the formula becomes:

• \(S = \frac{1.5}{1 - 0.01}\)

When simplified, it becomes \(S = \frac{1.5}{0.99}\), which simplifies further to approximately 1.51515....This sum represents how the series converges to a finite value, a feat achievable because the terms grow smaller due to the small common ratio.

Common Ratio Calculation

Finding the common ratio is a crucial step when working with geometric series, as it helps determine the behavior of the series.
The common ratio \(r\) is calculated by dividing a term in the series by the term preceding it. For our series:

• First term (\(a\)) = 1.5
• Second term = 0.015

The calculation is straightforward:

• \(r = \frac{0.015}{1.5} = 0.01\)

This calculation shows that each term is 0.01 times the previous term, effectively illustrating how the series shrinks rapidly. The smaller the common ratio, particularly when less than 1, the more likely the series will converge.

Infinite Series Formula

The infinite series formula is a handy tool that makes calculating the sum of a convergent geometric series effortless. The basic formula to remember is:

• \(S = \frac{a}{1 - r}\)

This formula leverages the concept of convergence. It accounts for how each successive term of the series contributes less to the sum.
The beauty of this mathematical shortcut is that as long as \( |r| < 1 \), you can substitute the first term \(a\) and the common ratio \(r\) into the formula to find the sum \(S\).
In the present example, substituting \(a = 1.5\) and \(r = 0.01\) simplifies the process immensely. The formula succinctly provides the sum of the infinite series as 1.51515..., revealing the power of mathematics in handling infinitely repeating processes.