Problem 22 - Calculus: Early Transcendentals

Question

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {5}{\pi^n} \)

Step-by-Step Solution

Answer

The series converges and its sum is \( \frac{5}{\pi - 1} \).

Step 1: Identify the First Term

The given series is \( \sum_{n=1}^{\infty} \frac{5}{\pi^n} \). The first term \( a \) of the series is the term when \( n = 1 \). This gives us \( a = \frac{5}{\pi} \).

Step 2: Determine the Common Ratio

The common ratio \( r \) of the series is the factor by which each term is multiplied to get the next term. Here, \( r = \frac{1}{\pi} \), as the ratio between successive terms \( \frac{5}{\pi^{n+1}} \) and \( \frac{5}{\pi^n} \) is \( \frac{1}{\pi} \).

Step 3: Check Convergence Condition

A geometric series \( \sum a r^{n-1} \) converges if and only if the absolute value of the common ratio \( |r| < 1 \). Here, \( |r| = \left| \frac{1}{\pi} \right| < 1 \) because \( \pi > 1 \). Therefore, the series converges.

Step 4: Compute the Sum of the Series

The sum \( S \) of an infinite convergent geometric series is given by the formula \( S = \frac{a}{1 - r} \). Substituting the values, we have: \[ S = \frac{\frac{5}{\pi}}{1 - \frac{1}{\pi}} = \frac{\frac{5}{\pi}}{\frac{\pi-1}{\pi}} = \frac{5}{\pi-1} \]. Thus, the sum of the series is \( \frac{5}{\pi-1} \).

Key Concepts

Convergence of Series

A geometric series is a series 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. Understanding whether such a series converges or diverges is key to working with it. Convergence in mathematics refers to the idea that as more terms of the series are added together, the sum approaches a fixed value. For a geometric series to be convergent, the absolute value of the common ratio \( r \) should be less than 1, that is, \( |r| < 1 \). This ensures that as you increase the number of terms in the series, the terms become smaller and smaller, leading the series towards a finite limit or sum.

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

In the provided example, \( r = \frac{1}{\pi} \). Since \( \pi \approx 3.14 \), it is clear that \( |\frac{1}{\pi}| < 1 \), confirming the series is convergent. This means the series will sum to a finite number, rather than grow indefinitely as you add more terms.

Common Ratio

The common ratio is a fundamental element of geometric series, which dictates how each term is derived from the previous one. It is denoted by \( r \). Specifically, in the formula for the series \( a, ar, ar^2, ar^3, \ldots \), each term is obtained by multiplying the previous term by \( r \).Understanding the value of \( r \) helps in analyzing how quickly or slowly the terms of the series grow.

• If \( r > 1 \), the terms will grow larger, leading to divergence.
• If \( r < 1 \), the terms will shrink, potentially leading to convergence.
• For this exercise, \( r = \frac{1}{\pi} \), indicating the terms will become smaller, pointing towards convergence.

The consistency offered by \( r \) is why geometric series can be so neatly managed, especially when \( |r| < 1 \), giving us the possibility to calculate a sum even if the series has infinitely many terms.

Sum of Infinite Series

Once we've established that a geometric series converges, we can actually find its sum, even if it includes infinitely many terms. This is possible because, under the right conditions, the series approaches a finite limit. The sum \( S \) of such a convergent infinite geometric series is found using the formula:\[S = \frac{a}{1 - r}\]where \( a \) is the first term, and \( r \) is the common ratio. The beauty of this formula lies in its simplicity; with just the first term and the common ratio, you can find the sum of an infinite series when \( |r| < 1 \). In the given exercise:

• The first term \( a = \frac{5}{\pi} \).
• The common ratio \( r = \frac{1}{\pi} \).

By substituting these into the formula, we find:\[S = \frac{\frac{5}{\pi}}{1 - \frac{1}{\pi}} = \frac{5}{\pi - 1}\]This demonstrates how a clear understanding of the components of a geometric series can simplify the process of finding the sum, even when dealing with infinity.