Problem 1 - Thomas Calculus

Question

In Exercises \(1-6,\) find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{3^{n-1}}+\cdots$$

Step-by-Step Solution

Answer

The series converges and its sum is 3.

Step 1: Identify the Series Type

The given series is \(2 + \frac{2}{3} + \frac{2}{9} + \frac{2}{27} + \cdots\). Observe that this is a geometric series where each term is a constant multiple of its predecessor. The first term \(a = 2\) and the common ratio \(r = \frac{1}{3}\).

Step 2: Determine the Formula for the n-th Partial Sum

For a geometric series with a first term \(a\) and common ratio \(|r| < 1\), the formula for the \(n\)-th partial sum \(S_n\) is given by: \[ S_n = a \frac{1 - r^n}{1 - r} \]. Substitute \(a = 2\) and \(r = \frac{1}{3}\) into the formula: \[ S_n = 2 \frac{1 - \left(\frac{1}{3}\right)^n}{1 - \frac{1}{3}} \].

Step 3: Simplify the Partial Sum Formula

Simplify the denominator: \(1 - \frac{1}{3} = \frac{2}{3}\). Therefore, the formula becomes: \[ S_n = 2 \cdot \frac{1 - \left(\frac{1}{3}\right)^n}{\frac{2}{3}} \]. Simplify further: \[ S_n = 3 \left(1 - \left(\frac{1}{3}\right)^n\right) \].

Step 4: Determine the Sum of the Infinite Series

The sum of an infinite geometric series \(S\) where \(|r| < 1\) is \( S = \frac{a}{1-r} \). For this series: \[ S = \frac{2}{1 - \frac{1}{3}} = 3 \]. Since the common ratio \(\left(\frac{1}{3}\right)\) is less than one, the series converges to this sum.

Key Concepts

Partial Sum

When working with a geometric series, understanding the concept of a partial sum can be incredibly useful. A partial sum is essentially the sum of the first few terms of a series. In mathematical terms, if you're given a series of the form \[S = a + ar + ar^2 + ar^3 + \dots \,\]the partial sum, denoted as \( S_n \), adds up the first \( n \) terms of this series.For a geometric series, calculating this involves using a specific formula. The formula for the \( n \)th partial sum \( S_n \) is:\[S_n = a \frac{1 - r^n}{1-r} \]Where \( a \) is the first term and \( r \) is the common ratio of the series. This formula allows us to quickly find the sum of any portion of the series.

• Choose the first term, \( a \).
• Identify the common ratio, \( r \).
• Apply these in the formula to get the partial sum \( S_n \).

By calculating the partial sums as the series progresses, we gain insights into the series' behavior and approach its total sum.

Convergence

In the context of series, convergence refers to the behavior of a series as the number of terms grows indefinitely. A series converges if the partial sums approach a finite limit. This is especially important when dealing with an infinite number of terms.For a geometric series, convergence depends on the value of the common ratio \( r \):

• If \( |r| < 1 \), the series will converge.
• If \( |r| \geq 1 \), the series will not converge.

The given series in our context has a common ratio of \( \frac{1}{3} \), which is less than one. Thus, it meets the criteria for convergence and the infinite series can be summed to a finite value.Convergence ensures that as you sum more and more terms, you get closer to a particular number. This makes convergence a key concept in understanding the behavior and limits of series.

Infinite Series

An infinite series is a sum of infinitely many terms. Unlike finite series, which stop after a certain number, infinite series continue indefinitely.Understanding infinite series begins with recognizing the pattern and behavior, such as a common ratio in geometric series. The sum of an infinite geometric series, where \( |r| < 1 \), can be calculated using the formula: \[ S = \frac{a}{1-r} \]Where \( a \) is the initial term and \( r \) is the common ratio.In our example series, substituting the values \( a = 2 \) and \( r = \frac{1}{3} \) into the formula yields:\[ S = \frac{2}{1 - \frac{1}{3}} = 3 \]This demonstrates that even though the series is infinite, the sum reaches a fixed value, known as the limit of the series. This understanding helps in evaluating not only geometric series but many other types of series that use similar concepts.