Problem 25 - Single Variable Calculus

Question

(a) compute as many terms of the sequence of partial sums, \(S_{n}\), as is necessary to convince yourself that the series converges or diverges. If it converges, estimate its sum. (b) Plot \(\left\\{S_{n}\right\\}\) to give a visual confirmation of your observation in part (a). (c). If the series converges, find the exact sum. If it diverges, prove it, using the Divergence Theorem. \(\sum_{n=1}^{\infty} 3\left(\frac{7}{8}\right)^{n-1}\)

Step-by-Step Solution

Answer

The given series is a geometric series with \(a_n = 3\left(\frac{7}{8}\right)^{n-1}\) and a common ratio of \(\frac{7}{8}\). Since \(|\frac{7}{8}| < 1\), the series converges. The sum of the series can be calculated using the formula for the sum of a geometric series, \(S = \frac{a_1}{1-r}\), which gives us \(S = \frac{3}{1-\frac{7}{8}} = 24\). A plot of the sequence of partial sums visually confirms the convergence of the series to a sum of 24.

Step 1: Identify the type of series

The given series is a geometric series with \(a_n = 3\left(\frac{7}{8}\right)^{n-1}\). Notice that the common ratio is \(\frac{7}{8}\). To determine the convergence of the series, we will use the formula for the sum of a geometric series. Step 2: Determine convergence or divergence

Step 2: Determine convergence or divergence

The convergence of a geometric series depends on the value of the common ratio. If \(|r| < 1\), the series converges, and if \(|r| \geq 1\), the series diverges. In this case, the common ratio is \(r =\frac{7}{8}\). Since $|r| = |\frac{7}{8}| < 1, the series converges. Step 3: Calculate the sum of the series (if it converges)

Step 3: Calculate the sum of the series

Since we determined that the series converges, we can now calculate its sum using the formula for the sum of a geometric series: \(S = \frac{a_1}{1-r}\), where \(a_1\) is the first term and \(r\) is the common ratio. In this case, \(a_1 = 3\) and \(r = \frac{7}{8}\). Evaluate the sum as follows: \[S = \frac{3}{1-\frac{7}{8}} = \frac{3}{\frac{1}{8}} = 3 \cdot 8 = 24\] Step 4: Visual confirmation of convergence

Step 4: Visual confirmation of convergence

To visually confirm the convergence of the series, we need to plot the sequence of partial sums \(S_n\). As previously determined, the series converges to a sum of 24. When plotting the sequence of partial sums, you should observe that the values approach 24 as n increases. You can use a graphing calculator or software like Desmos to plot the sequence of partial sums and observe the convergence. Step 5: Conclusion

Step 5: Conclusion

We have determined that the given series converges because the absolute value of the common ratio is less than 1. We then calculated the sum of the series, which is 24, and can visually confirm this value by plotting the sequence of partial sums.

Key Concepts

Convergence

Convergence is a key concept in the study of series in mathematics, specifically when determining whether a given series will settle towards a particular value. For a geometric series, the convergence is primarily dependent on the common ratio, denoted as \( r \). Here's a simple rule:

• If the absolute value of \( r \) is less than 1, \(|r| < 1\), the series converges. This means that as you add more terms of the series, the total sum approaches a finite number.
• If the absolute value of \( r \) is equal to or greater than 1, \(|r| \geq 1\), the series diverges, indicating the sum continues to grow without bound.

In our example, the common ratio is \( \frac{7}{8} \), which means \(|\frac{7}{8}| < 1\). Therefore, the series converges, implying it will approach a particular sum as more terms are added to the series.

Partial Sums

The concept of partial sums is crucial in understanding series and their behavior. A partial sum, denoted as \( S_n \), represents the sum of the first \( n \) terms of a series. By examining

• the sequence of partial sums, one can gain insight into whether the series converges or diverges.

In practice, we calculate a number of partial sums and observe the pattern they form. If the partial sums begin to stabilize or settle towards a single value, this suggests convergence. In the original exercise, plotting partial sums helps visually confirm whether the series converges towards the sum of 24.

Sum of a Series

The sum of a series refers to the value that the infinite addition of terms approaches if the series converges. For a geometric series, the sum \( S \) can be calculated with a precise formula:\[ S = \frac{a_1}{1-r} \]where:

• \( a_1 \) is the first term of the series, and
• \( r \) is the common ratio.

In the exercise, the first term \( a_1 \) is 3, and the common ratio \( r \) is \( \frac{7}{8} \). Plugging these into the formula gives us:\[ S = \frac{3}{1 - \frac{7}{8}} = \frac{3}{\frac{1}{8}} = 3 \times 8 = 24 \]Hence, the series converges, and its sum is 24.

Divergence Theorem

The Divergence Theorem is a fundamental concept that deals with the behavior of series beyond geometric ones. It asserts that if the terms of a series do not approach zero, the series cannot converge. However, it's more common in calculus related to vector fields. For series, our interest is rather in stating that if the partial sums do not approach a finite value, divergence occurs.In the context of the exercise presented, the Divergence Theorem would be used to justify the absence of convergence when the criteria for a convergent series are not met. However, for our geometric series example, convergence holds because the common ratio condition \(|r| < 1\) is satisfied, so the Divergence Theorem's assertion of divergence is not needed. This makes geometric series simply reliant on the common ratio and its magnitude compared to 1 for determining convergence or divergence.