Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Do the following. i. Write out the rst two terms of the series. ii. Determine whether or not the series converges. iii. If the series converges, determine its sum. $$ \sum_{n=3}^{\infty} \frac{(-1)^{n} 3}{2^{n}} $$

Step-by-Step Solution

Answer

The first two terms of the series are \(-\frac{3}{8}\) and \(\frac{3}{16}\). The series converges because the absolute value of the common ratio is less than 1. The sum of this infinite series is \(-1\).

Step 1: Identify the First Two Terms

To find the first two terms of the series, substitute \(n = 3\) and \(n = 4\) into \(\frac{(-1)^{n} 3}{2^{n}}\). This yields \(-\frac{3}{8}\) and \(\frac{3}{16}\) respectively.

Step 2: Convergence of the Series

For the series to converge, the magnitude of the common ratio must be less than 1. The common ratio \(\frac{-1}{2}\), which is identified as the expression multiplied by each subsequent term to reach the next one, has magnitude less than 1 and hence the series converges.

Step 3: Calculate the Sum of the Series

If the series converges, its sum can be ascertained using the formula \(S = \frac{a}{1 - r}\). With \(a = -\frac{3}{8}\) and \(r = \frac{-1}{2}\), the sum of the series equals \(-\frac{3}{8} / (1 + \frac{1}{2}) = -\frac{3}{3} = -1\).

Key Concepts

Convergence

In mathematics, convergence of a series refers to the behavior of a series as the number of terms extends to infinity. For a series to converge, it must approach a specific value, known as the limit. This is a key concept because it helps us determine whether the series yields a finite sum or not, making it a crucial part of mathematical analysis.

In the context of our exercise, we assessed convergence by considering the common ratio of the series, \(r = -\frac{1}{2}\). This value is less than 1 in absolute terms, meaning the series converges.

• When \(|r| < 1\), the terms of the series become smaller and approach zero as \(n \) becomes very large.
• If \(|r| \geq 1\), the series diverges and does not converge to a limit.

This check on the common ratio is a staple method for examining the convergence of geometric series.

Geometric Series

A geometric series is a series with a constant ratio between successive terms. Recognizing this type of series is important for applying specific convergence tests and formulas. Its defining characteristic is the common ratio, \(/ r /\), which importantly determines the behavior of the series.

Our exercise gives us a geometric series, since we can express the given series as:
\(a_n = \frac{(-1)^n 3}{2^n}\)

The common ratio, in this case, is \(r = -\frac{1}{2}\). This means each term in the series after the first is obtained by multiplying the preceding term by \(-\frac{1}{2}\).

• This multiplicative pattern defines the series as geometric.
• Geometric series are uniquely simple, allowing easy application of convergence and summation rules given the value of \(r\).

Identifying the common ratio quickly informs us about convergence and forms the basis for finding the sum if it converges.

Sum of Series

Calculating the sum of a convergent infinite geometric series hinges on using a particular formula that capitalizes on the common ratio. The sum \(S\) of such series can be determined as long as the series converges, using the formula:\[ S = \frac{a}{1 - r} \]

Here \(a\) is the first term of the series, and \(r\) is the common ratio. This is what we apply to conclude that the sum exists and is finite. For our series:
\(a = -\frac{3}{8}\)
\(r = -\frac{1}{2}\)

By substituting these values into the formula, the sum \(S\) is calculated as:
\[ S = \frac{-\frac{3}{8}}{1 - (-\frac{1}{2})} = \frac{-\frac{3}{8}}{\frac{3}{2}} = -1 \]

Thus, the sum of the series is \(-1\).

• This formula is key for solving any convergent infinite geometric series.
• It highlights the simplicity and necessity of recognizing geometric behavior in series to reach solutions quickly.