Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series $$ \sum_{k=1}^{\infty}\left(\frac{e}{\pi}\right)^{k+1} $$

Step-by-Step Solution

Answer

The series converges with sum \( \frac{e^2}{\pi(\pi - e)} \).

Step 1: Identify the Series Type

The series given is \( \sum_{k=1}^{\infty}\left(\frac{e}{\pi}\right)^{k+1} \). This is a geometric series since each term has the form \( ar^{k} \), where \( a \) is the first term, and \( r \) is the common ratio. Here, \( r = \frac{e}{\pi} \).

Step 2: Determine Convergence

A geometric series \( \sum ar^k \) converges if the absolute value of the common ratio \( |r| < 1 \). Here, \( |r| = \left|\frac{e}{\pi}\right| \). Since \( e \approx 2.718 \) and \( \pi \approx 3.1416 \), it is clear that \( \frac{e}{\pi} < 1 \). Hence, the series converges.

Step 3: Find the First Term

To find the first term of the series, calculate \( a = r \) at \( k=1 \). This gives us \( a = \left(\frac{e}{\pi}\right)^{1+1} = \left(\frac{e}{\pi}\right)^{2} \).

Step 4: Calculate the Sum of the Series

The sum of an infinite geometric series is given by the formula \( S = \frac{a}{1-r} \). Substitute \( a = \left(\frac{e}{\pi}\right)^2 \) and \( r = \frac{e}{\pi} \) into the formula:\[S = \frac{\left(\frac{e}{\pi}\right)^2}{1 - \frac{e}{\pi}}\]

Step 5: Simplify the Expression

To simplify, compute the expression:\[1 - \frac{e}{\pi} = \frac{\pi - e}{\pi}\]Substitute back into the sum formula:\[S = \frac{\left(\frac{e}{\pi}\right)^2}{\frac{\pi - e}{\pi}} = \frac{e^2}{\pi(\pi - e)}\]

Key Concepts

Convergence and Divergence of Series

When dealing with infinite series, one primary goal is to determine whether the series converges or diverges. A series converges if the sum of its infinite terms approaches a finite number as more terms are added. Conversely, a series diverges if the sum does not settle to a definite value.

With geometric series, the key to determining convergence lies in the common ratio. The standard form of a geometric series is \( \sum ar^k \), where \( a \) is the first term and \( r \) is the common ratio.

• If the absolute value of the common ratio \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series diverges.

In our original exercise, the common ratio was \( \frac{e}{\pi} \), approximately 0.866. Since this value is less than 1, the series converges. Understanding the concept of convergence can help predict whether the sum is finite without computation.

Sum of Infinite Series

Finding the sum of an infinite series is possible using specific formulas, especially for geometric series. Once we know a geometric series converges, we can calculate its sum using the formula:

• \( S = \frac{a}{1-r} \)

Here, \( a \) is the first term of the series, and \( r \) is the common ratio. This formula works only when \( |r| < 1 \), ensuring that the series converges to a finite sum.

In the exercise provided, the first term was determined as \( a = \left(\frac{e}{\pi}\right)^2 \), and \( r = \frac{e}{\pi} \). Substituting these values into the formula allows us to calculate the sum of the series. This step-by-step method simplifies the process, demonstrating how the sum represents all the terms added to infinity.

Geometric Series Formula

The geometric series formula is crucial for understanding and solving series involving a constant ratio between consecutive terms. The general form of a geometric series can be written as:

• \( \sum_{k=0}^{\infty} ar^k \)

Where:

• \( a \) is the first term
• \( r \) is the common ratio

This formula is especially powerful for solving real-world problems where patterns or sequences repeat, such as in finance for calculating interest or in physics for wave patterns.

In mathematical exercises like your original problem, identifying the type of series and correctly applying the geometric series formula streamlines solving for convergence, divergence, and the sum. By plugging in the first term and ratio into the sum formula, you uncover insights into how seemingly endless series can encapsulate whole concepts into a single finite number.