Problem 11 - Essential Calculus Early Transcendentals

Question

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=0}^{\infty} \frac{\pi^{n}}{3^{n+1}}$$

Step-by-Step Solution

Answer

The series is divergent, it does not converge.

Step 1: Identify the series form

The given series is a geometric series. A geometric series has the form \( \sum_{n=0}^{\infty} ar^n \). Our series is \( \sum_{n=0}^{\infty} \frac{\pi^{n}}{3^{n+1}} \). To match it to the geometric series form, we rewrite it as \( \sum_{n=0}^{\infty} \left(\frac{1}{3}\right) \left(\frac{\pi}{3}\right)^n \). Here, \( a = \frac{1}{3} \) and \( r = \frac{\pi}{3} \).

Step 2: Check for convergence condition

A geometric series converges if and only if the absolute value of its common ratio \( |r| < 1 \). For our series, \( r = \frac{\pi}{3} \). Since \( \pi \approx 3.14159 \), we find \( \left| \frac{\pi}{3} \right| = \frac{\pi}{3} > 1 \).

Step 3: Determine convergence

Since the common ratio \( \left| \frac{\pi}{3} \right| > 1 \), the series does not satisfy the convergence condition \( |r| < 1 \). Therefore, the series is divergent.

Key Concepts

Series Convergence

When we talk about series convergence in the context of a geometric series, we're discussing whether the sum of the series approaches a fixed number as more and more terms are added. Convergence is a critical concept because it tells us if we can actually calculate a definite, finite sum from an infinite sequence. For geometric series, the main factor that determines convergence is the common ratio, denoted as \( r \).

If the absolute value of the common ratio \( |r| \) is less than 1, the series converges, and there is a formula to find its sum: \[S = \frac{a}{1 - r}\]where \( a \) is the first term of the series.

• Convergence means getting closer to a fixed number.
• A common ratio, \( |r| \), less than 1 is necessary for convergence.
• An infinite series sum can be finite if it converges.

In our given series, we first identify whether it converges by checking this value of the common ratio.

Common Ratio

The common ratio, denoted \( r \), is the factor by which each term of a geometric series is multiplied to get the next term. Identifying \( r \) is essential because it directly influences whether the series converges or diverges. Formula for the common ratio in a geometric series is represented as:\[ar^n\]Thus, finding the ratio is necessary to determine the behavior of the series. In the series example provided:

• We rewrite the series: \( \sum_{n=0}^{\infty} \frac{1}{3} \left(\frac{\pi}{3}\right)^n \).
• Here, \( a = \frac{1}{3} \) and \( r = \frac{\pi}{3} \).

Understanding the common ratio helps us establish the next steps for analysis, whether to find a sum or declare divergence.

Divergent Series

A divergent series is one where as you add more terms, the series does not approach a finite limit. Instead, the sum of its terms heads towards infinity or doesn't settle at any particular value. This lack of convergence can be easily identified in a geometric series by examining its common ratio. If \( |r| \geq 1 \), the series is classified as divergent.

In our specific series:

• The common ratio is \( r = \frac{\pi}{3} \), approximately equal to \( 1.047 \).
• Since \( \frac{\pi}{3} > 1 \), the condition for convergence, \( |r| < 1 \), isn't met.

Therefore, the series does not converge to a finite value and is considered divergent. Understanding divergence and convergence is crucial for determining the behavior of infinite series and whether meaningful sums can be obtained.