Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}+\cdots $$

Step-by-Step Solution

Answer

The series converges and its sum is \(2/3\).

Step 1: Identify Series Type

1. Identify the type of the series. In this case, the series is a geometric series, which is represented by the summation \(a+ar+ar^2+ar^3+...\) where \(a\) is the first term and \(r\) is the common ratio between terms.

Step 2: Find Common Ratio

2. Identify the common ratio. The common ratio \(r\) for the given geometric series \(1 -1/2 + 1/4 - 1/8 + 1/16 +...\) is \(r = -1/2\), found by dividing the second term by the first term (-1/2 divided by 1).

Step 3: Check Absolute Value of Common Ratio

3. Determine if the geometric series converges by checking if the absolute value of the common ratio is less than 1. In this case, the absolute value of the common ratio \(-1/2\) is \(| -1/2 |= 1/2\), which indeed less than 1.

Step 4: Find Sum of Convergent Series

4. If the series converges, find its sum. The sum \(s\) of a geometric series is given by the formula \(s= a/(1-r)\). Here, \(a = 1\) and \(r = - 1/2\), so \(s = 1/(1 - (-1/2)) = 2/3 \)

Key Concepts

Geometric Series

A geometric series is a fascinating type of series in mathematics where each term is simply multiplied by a constant known as the *common ratio* to produce the next term. Recognizing a geometric series is crucial because it helps us determine whether a series will converge to a single value or diverge. The general form of a geometric series is:

• a + ar + ar^2 + ar^3 + ...

where:

• a is the first term of the series.
• r is known as the common ratio.

In our given series, for instance, it was identified as geometric because each term after the first is derived by multiplying the previous term by the same constant, which is the common ratio. Once a series is identified as geometric, we gain a powerful tool for determining if the series reaches a finite sum or grows indefinitely.

Common Ratio

The *common ratio* in a geometric series is the factor by which each term is multiplied to obtain the subsequent term. It is a key component in determining the behavior of the series. To find this ratio, simply divide any term by the preceding term in the series.In the series:

• 1, -1/2, 1/4, -1/8, ... ,

the common ratio is calculated by dividing the second term by the first, giving us \[ r = \frac{-1/2}{1} = -1/2 \].Determining the absolute value of the common ratio is crucial because:

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

For our series, the common ratio's absolute value is \( | -1/2 | = 1/2 \), which is less than 1, indicating convergence towards a finite sum.

Sum of Series

When a geometric series converges, we can find its sum using a straightforward formula. The sum, denoted by \( S \), of an infinite geometric series where the common ratio's absolute value is less than 1 is:\[S = \frac{a}{1 - r}\]Here, \( a \) is the first term of the series, and \( r \) is the common ratio.
Let's apply this in the context of our specific series:

• First term \( a = 1\)
• Common ratio \( r = -1/2 \)

Using the sum formula:\[S = \frac{1}{1 - (-1/2)} = \frac{1}{1 + 1/2} = \frac{1}{3/2} = \frac{2}{3}\]As a result, the sum of our geometric series is \( \frac{2}{3} \).
This demonstrates why understanding the properties of a geometric series and its common ratio enables us to calculate actual sums of infinite series—a powerful concept in infinite series analysis.