Problem 22 - Essential Calculus Early Transcendentals

Question

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{k=1}^{\infty}(\cos 1)^{k}$$

Step-by-Step Solution

Answer

The series is convergent with sum approximately 1.175.

Step 1: Recognize the series type

The series given is \( \sum_{k=1}^{\infty}(\cos 1)^{k} \). Since each term is of the form \((\cos 1)^k\), where \(\cos 1\) is a constant, this is a geometric series with the first term \(a = (\cos 1)^1 = \cos 1\) and common ratio \(r = \cos 1\).

Step 2: Check the convergence criteria for a geometric series

A geometric series \(\sum_{k=0}^{\infty} ar^k\) is convergent if and only if \(|r| < 1\). Here, \(|r| = |\cos 1| < 1\) since \(\cos 1 \approx 0.5403\), which is less than 1. Hence, the series is convergent.

Step 3: Use the formula for the sum of a convergent geometric series

The sum \(S\) of an infinite geometric series with first term \(a\) and common ratio \(|r| < 1\) is given by the formula \(S = \frac{a}{1-r}\). Here, \(a = \cos 1\) and \(r = \cos 1\), so the sum is \(S = \frac{\cos 1}{1 - \cos 1}\).

Step 4: Calculate the value of the sum

Substitute the approximate value of \(\cos 1 \approx 0.5403\) into the formula to find the sum: \[ S = \frac{0.5403}{1 - 0.5403} \approx \frac{0.5403}{0.4597} \approx 1.175 \]. So, the sum is approximately 1.175.

Key Concepts

Geometric Series Convergence

A geometric series occurs when each term in the series is derived by multiplying the previous term by a constant, known as the common ratio \( r \).
If we express it formally:

• The first term of the series is represented as \( a \).
• Each subsequent term is \( ar, ar^2, ar^3, \) and so on. This forms an infinite geometric series \( \sum_{k=0}^{\infty} ar^k \).

The crucial aspect for understanding geometric series is determining if they converge, meaning they approach a finite sum as more terms are added.
The convergence of a geometric series is strictly determined by its common ratio \( r \).

• A series will converge if the absolute value of the common ratio is \( |r| < 1 \).
• If \( |r| \geq 1 \), the series diverges and does not sum to a finite number.

In our exercise, since \( \cos 1 \) approximately equals 0.5403 and \( |\cos 1| < 1 \), the series \( \sum_{k=1}^{\infty} (\cos 1)^k \) converges.

Infinite Series Sum

Once it is determined that a geometric series converges, we can calculate its sum. This applies only to infinite series where the ratio \( |r| < 1 \).
The sum of the series can be calculated using the formula:

• \( S = \frac{a}{1-r} \) where \( a \) represents the first term and \( r \) is the common ratio.

To solve our given example, let us specify the values:

• \( a = \cos 1 \) is the first term
• \( r = \cos 1 \) is also the common ratio

Substituting these values into the formula, we have:\[ S = \frac{\cos 1}{1 - \cos 1} \]Using the approximation \( \cos 1 \approx 0.5403 \):\[ S \approx \frac{0.5403}{0.4597} \approx 1.175 \]So, for our series, the sum is approximately 1.175, confirming that it indeed converges to a finite value.

Series Convergence Criteria

The convergence criteria for any series, including geometric ones, is a set of conditions that allow us to determine if the series approaches a specific value as more terms are added.
For geometric series, the convergence criterion is simple and crucial:

• If the absolute value of the common ratio \( |r| \) is less than 1, the series converges.
• Conversely, if \( |r| \geq 1 \), the series is divergent.

In the case of our geometric series \( \sum_{k=1}^{\infty} (\cos 1)^k \):

• The value of \( \cos 1 \approx 0.5403 \), which is within the convergent range \(|r| < 1\).
• This fulfillment of the convergence criterion ensures the series adds up to a calculable finite number.

Understanding these criteria confirms why and how certain series achieve convergence, a fundamental part of summarizing infinite processes in mathematics.