Problem 21 - Thomas Calculus

Question

In Exercises \(15-22\) , determine if the geometric series converges or diverges. If a series converges, find its sum. $$1+\left(\frac{10}{9}\right)^{2}+\left(\frac{10}{9}\right)^{4}+\left(\frac{10}{9}\right)^{6}+\left(\frac{10}{9}\right)^{8}+\cdots$$

Step-by-Step Solution

Answer

The series diverges; it does not have a sum.

Step 1: Recognize the Series Form

The given series is of the form: \( 1 + \left(\frac{10}{9}\right)^2 + \left(\frac{10}{9}\right)^4 + \left(\frac{10}{9}\right)^6 + \cdots \). This is a geometric series because each term is obtained by multiplying the previous term by a constant ratio.

Step 2: Identify the First Term and Common Ratio

In a geometric series, the first term (\( a \)) is the initial term, and the common ratio (\( r \)) is the factor by which we multiply the previous term to get the next term. Here, \( a = 1 \) and \( r = \left(\frac{10}{9}\right)^2 \).

Step 3: Determine the Convergence

A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). In this case, \( r = \left(\frac{10}{9}\right)^2 = \frac{100}{81} \). Since \( \left| \frac{100}{81} \right| > 1 \), the series diverges.

Step 4: Conclusion for Diverging Series

Since the series diverges, it does not have a finite sum. For a geometric series to converge, the common ratio must be less than 1 in absolute value, which is not the case here.

Key Concepts

Series Convergence

Understanding whether a series converges or not is crucial in analyzing geometric series. A geometric series converges when the absolute value of its common ratio, denoted as \( |r| \), is less than 1. This condition ensures that as the number of terms increases, the series approaches a specific finite value.
If a geometric series converges, the sum \( S \) of the infinite series can be found using the formula:

• \( S = \frac{a}{1-r} \)

where \( a \) is the first term and \( r \) is the common ratio.
Convergence is key because it tells us whether we can assign a definite value to an infinite series. Without it, we'd deal with sums that could grow without bounds.

Common Ratio

In a geometric series, the common ratio \( r \) is the factor multiplied by each term to obtain the next term in the series. For instance, in the series \( 1 + \left(\frac{10}{9}\right)^2 + \left(\frac{10}{9}\right)^4 + \cdots \), the common ratio is \( \left(\frac{10}{9}\right)^2 \).
Identifying the common ratio is essential as it determines the behavior of the series.

• If \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series diverges.

In the given example, \( r = \frac{100}{81} \), meaning the series diverges because \( \left| \frac{100}{81} \right| > 1 \). The common ratio acts as an indicator of the series' overall behavior in terms of convergence and divergence.

Divergence

When a geometric series diverges, it means that the series does not approach a finite sum. Rather than settling towards a particular value, the series continues to grow indefinitely.
Divergence occurs when the absolute value of the common ratio \( |r| \) is equal to or greater than 1. For instance, in our example series, the common ratio \( r = \left(\frac{100}{81}\right) \) is greater than 1, thus indicating divergence.
Divergent series are less useful in scenarios where a finite sum is required, as they essentially lack mathematical end-definition.

• This scenario means calculations involving infinite terms result in outputs that cannot be quantified into a practical value.

Infinite Series

Infinite series are essentially sums of an infinite sequence of numbers. The geometric series is a type of infinite series where each term is derived by multiplying the previous term by a constant factor known as the common ratio.
Infinite series are powerful tools in mathematics because they allow for representation of complex functions through simpler terms.

• Convergent infinite series can represent finite values, such as in calculus or physics.
• Divergent ones, like our example, provide insights but are not used for finite solutions.

Understanding the behavior of infinite series through convergence, divergence, and common ratios is vital for applying mathematical concepts practically. They serve as the building blocks for further mathematical innovation and problem-solving.