Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=0}^{\infty}\left[2\left(\frac{1}{4}\right)^{k}+3\left(-\frac{1}{5}\right)^{k}\right] $$

Step-by-Step Solution

Answer

The series converges, and its sum is \( \frac{31}{6} \).

Step 1: Identify the series components

The given series is \( \sum_{k=0}^{\infty}\left[2\left(\frac{1}{4}\right)^{k}+3\left(-\frac{1}{5}\right)^{k}\right] \). This is a sum of two separate geometric series: 1. \( 2\left(\frac{1}{4}\right)^{k} \)2. \( 3\left(-\frac{1}{5}\right)^{k} \). We'll address each series separately to determine their convergence and sum.

Step 2: Analyze the first series

The first series is \( \sum_{k=0}^{\infty}2\left(\frac{1}{4}\right)^{k} \). It's a geometric series with first term \( a = 2 \) and common ratio \( r = \frac{1}{4} \). Since \(|r| = \frac{1}{4} < 1\), the series converges. The sum of an infinite geometric series is given by \( \frac{a}{1-r} \). Therefore, the sum of the first series is: \( \frac{2}{1-\frac{1}{4}} = \frac{2}{\frac{3}{4}} = \frac{8}{3} \).

Step 3: Analyze the second series

The second series is \( \sum_{k=0}^{\infty}3\left(-\frac{1}{5}\right)^{k} \). It's a geometric series with first term \( a = 3 \) and common ratio \( r = -\frac{1}{5} \). Since \(|r| = \frac{1}{5} < 1\), this series also converges. Its sum is given by \( \frac{a}{1-r} \). Therefore, the sum of the second series is: \( \frac{3}{1-(-\frac{1}{5})} = \frac{3}{1 + \frac{1}{5}} = \frac{3}{\frac{6}{5}} = \frac{15}{6} = \frac{5}{2} \).

Step 4: Determine the total sum

Since both series converge, the sum of the original series is the sum of the two series we calculated:\( \sum_{k=0}^{\infty}\left[2\left(\frac{1}{4}\right)^{k}+3\left(-\frac{1}{5}\right)^{k}\right] = \frac{8}{3} + \frac{5}{2} \). To add these fractions, we find a common denominator. The least common denominator of 3 and 2 is 6:\( \frac{8}{3} = \frac{16}{6} \)\( \frac{5}{2} = \frac{15}{6} \)Therefore, the total sum is:\( \frac{16}{6} + \frac{15}{6} = \frac{31}{6} \).

Key Concepts

Geometric Series

A geometric series is a sequence 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. This makes these series incredibly useful for calculating the sum of complex patterns that follow such a progression.
For example, the sequence of terms in a geometric series can look like this:

• First term: \( a \)
• Second term: \( ar \)
• Third term: \( ar^2 \)
• And so forth: \( ar^3, ar^4, \, ... \)

A key aspect of geometric series is determining whether they converge or diverge. A geometric series converges when its common ratio \( r \) has an absolute value less than 1 (i.e., \( |r| < 1 \)). If a geometric series converges, you can find its sum using the formula:
\[ S = \frac{a}{1-r} \]
where \( a \) is the first term and \( r \) is the common ratio.

Infinite Series

An infinite series extends indefinitely, adding terms forever. It sounds intimidating, but in mathematics, we often investigate whether these series can settle into a single, finite number. This process is called convergence.
The series in the exercise was an infinite series, comprised of the combination of two separate geometric series. Each part involved an infinite number of terms, shown by the notation \( \sum_{k=0}^{\infty} \), indicating that the sum starts at \( k = 0 \) and continues infinitely.
To determine if these infinite series converge, we rely on the criterion that the absolute value of the common ratio must be less than one. If so, even though terms add to infinity, their sum approaches a finite number. This exercise exemplifies how sometimes, the behavior of infinite series is easier to measure by breaking them down into manageable parts.

Common Ratio

The common ratio is a constant that defines the progression of a geometric series. It is the factor by which we multiply each term to get to the next one. This characteristic is fundamental in recognizing and working with geometric series.
In the exercise, two different common ratios were presented:

• The first series had a common ratio of \( \frac{1}{4} \)
• The second series had a common ratio of \( -\frac{1}{5} \)

Identifying these ratios is crucial because it determines whether each series converges or diverges. In both cases, since the absolute values \( \left| \frac{1}{4} \right| = 0.25 \) and \( \left| -\frac{1}{5} \right| = 0.2 \) are less than 1, we found that each geometric series converged.
Understanding the common ratio is pivotal when computing the sum of an infinite geometric series. It tells us not only about the series' convergence but also plays a role in determining its sum using the sum formula \( \frac{a}{1-r} \).