Problem 34 - Calculus: Early Transcendentals

Question

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {2^n + 4^n}{e^n} \)

Step-by-Step Solution

Answer

The series is convergent; total sum is \( \frac{2}{e-2} + \frac{4}{e-4} \).\)

Step 1: Simplify the Series Expression

Start by separating the terms in the numerator: \[ \sum_{n=1}^{\infty} \frac{2^n}{e^n} + \sum_{n=1}^{\infty} \frac{4^n}{e^n} \] This breaks down the original series into two separate geometric series.

Step 2: Identify First Term and Ratio for Each Series

For the first series \( \sum_{n=1}^{\infty} \left(\frac{2}{e}\right)^n \):- The first term \( a_1 = \frac{2}{e} \)- The common ratio \( r_1 = \frac{2}{e} \)For the second series \( \sum_{n=1}^{\infty} \left(\frac{4}{e}\right)^n \):- The first term \( a_2 = \frac{4}{e} \)- The common ratio \( r_2 = \frac{4}{e} \)

Step 3: Determine Convergence by Checking the Common Ratio

A geometric series \( \sum_{n=0}^{\infty} ar^n \) converges if \(|r| < 1\).- \( r_1 = \frac{2}{e} \) and since \( e > 2 \), \(|r_1| < 1\), so the series \( \sum_{n=1}^{\infty} \left(\frac{2}{e}\right)^n \) converges.- \( r_2 = \frac{4}{e} \) and since \( e \approx 2.718 \), \(|r_2| < 1\) as \( \frac{4}{2.718} < 1 \), so the series \( \sum_{n=1}^{\infty} \left(\frac{4}{e}\right)^n \) also converges.

Step 4: Calculate the Sum for Each Convergent Series

The sum \( S \) of an infinite geometric series is given by \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.- For the first series, \( S_1 = \frac{\frac{2}{e}}{1 - \frac{2}{e}} = \frac{2}{e - 2} \).- For the second series, \( S_2 = \frac{\frac{4}{e}}{1 - \frac{4}{e}} = \frac{4}{e - 4} \).

Step 5: Find the Total Sum of Both Series

Add the sums of the two series to get the total sum of the original series:\[ S_{total} = S_1 + S_2 = \frac{2}{e - 2} + \frac{4}{e - 4} \]

Key Concepts

Geometric Series

A geometric series is a type of series where each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. This is why geometric series are often easily identified by a recurring multiplication pattern.

An example of a simple geometric series is:

• 3, 6, 12, 24, ...

Here, the first term is 3, and the common ratio is 2 since every term is twice the previous one.

• First term, denoted as \(a\), is the starting point of the series.
• Common ratio \(r\) is the constant multiplying factor.

Geometric series can either converge or diverge, depending on the absolute value of the common ratio. Understanding the behavior of geometric series plays a crucial role in solving complex series problems.

Infinite Series

An infinite series is a series that continues indefinitely without an end. Such series are written with a summation symbol \(\sum\) expressing that the terms extend to infinity.

One common type of infinite series is where terms gradually approach zero as the series progresses, potentially converging to a finite sum.

• Written as \(\sum_{n=0}^{\infty} a_n\) where \(a_n\) is each term in the series.
• Can converge to a finite sum or diverge.

Infinite series are especially interesting in mathematics and can model natural phenomena, mathematical functions, or even economic principles, raising their significance in various applications.

Common Ratio

The common ratio in a geometric series is a vital factor that determines the nature of the series. It is the factor by which we multiply each term to get the next term in the series. For a series to be geometric, this ratio must be constant throughout.

• In a series like 2, 4, 8, ..., the common ratio \(r\) is 2.
• Consistent application of \(r\) generates subsequent terms.

Furthermore, the common ratio is essential in deciding whether the series converges or diverges. Specifically, in infinite geometric series, if the absolute value \(|r| < 1\), the series converges, otherwise, it diverges.

Convergent Series

A convergent series is one whose terms approach a specific limit, allowing it to sum to a particular finite value. Understanding convergence is crucial for determining series behavior, especially when infinite terms are involved.

For geometric series, convergence is dictated by the common ratio:

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

When a geometric series converges, the sum \(S\) is calculated using the formula:\[S = \frac{a}{1-r}\]where \(a\) is the first term, and \(r\) is the common ratio.

• Helps in determining the sum of temperature series, financial forecast models, etc.

In practical applications, convergent series are beneficial for approximating complex values.