Problem 45 - Essential Calculus Early Transcendentals

Question

Find the value of $$c\( if \)\sum_{n=2}^{\infty}(1+c)^{-n}=2$$

Step-by-Step Solution

Answer

The value of \(c\) is \(\frac{-1 + \sqrt{3}}{2}\).

Step 1: Understand the Given Series

We need to find the value of \(c\) such that the series \(\sum_{n=2}^{\infty}(1+c)^{-n}\) equals 2. This is an infinite geometric series starting at \(n=2\).

Step 2: Write Down the First Term and Common Ratio

The series given is \((1+c)^{-2}, (1+c)^{-3}, (1+c)^{-4}, \ldots\). This is a geometric series where the first term \(a = (1+c)^{-2}\) and the common ratio \(r = \frac{1}{1+c}\).

Step 3: Use the Formula for Infinite Geometric Series

For an infinite series \(\sum_{n=0}^{\infty} ar^n\), the sum is \(\frac{a}{1-r}\) provided \(|r| < 1\). In our series, \(a = (1+c)^{-2}\) and \(r = \frac{1}{1+c}\), so the sum formula becomes: \[ \frac{(1+c)^{-2}}{1 - \frac{1}{1+c}} = 2 \]

Step 4: Simplify the Sum Formula

First simplify the denominator: \[ 1 - \frac{1}{1+c} = \frac{1+c-1}{1+c} = \frac{c}{1+c} \] Plug it back into the formula: \[ \frac{(1+c)^{-2}}{\frac{c}{1+c}} = 2 \]

Step 5: Solve for \(c\)

Multiply both sides by \(\frac{c}{1+c}\): \[(1+c)^{-2} \cdot \frac{1+c}{c} = 2\] Simplify: \[\frac{1}{c(1+c)} = 2\] Cross-multiply and solve for \(c\): \[1 = 2c(1+c)\] \[1 = 2c + 2c^2\] Rearrange into a quadratic equation: \[2c^2 + 2c - 1 = 0\]

Step 6: Apply the Quadratic Formula

Use the quadratic formula \(c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 2\), \(b = 2\), and \(c = -1\): \[c = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 2 \cdot (-1)}}{4}\]\[c = \frac{-2 \pm \sqrt{4 + 8}}{4}\]\[c = \frac{-2 \pm \sqrt{12}}{4}\]\[c = \frac{-2 \pm 2\sqrt{3}}{4}\]\[c = \frac{-1 \pm \sqrt{3}}{2}\]

Step 7: Select the Valid Solution

Since \(c\) must be such that \(|r| < 1\), this means \(0 < 1+c < 2\), resulting in \(-1 < c < 1\). Therefore, the valid value for \(c\) is \(c = \frac{-1 + \sqrt{3}}{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \[ ax^2 + bx + c = 0. \]This formula provides the solutions (or roots) of the quadratic equation, given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]Here's a breakdown:

• a, b, c are coefficients from the quadratic equation, where \(a eq 0\).
• The term under the square root, \(b^2 - 4ac\), is called the discriminant. It determines the nature of the roots.
• The solutions can be real and distinct, real and repeated, or complex, depending on the value of the discriminant.

In the problem provided, the quadratic formula is used to solve the equation \(2c^2 + 2c - 1 = 0\). By identifying \(a = 2\), \(b = 2\), and \(c = -1\), we input these into the formula to find the possible values of \(c\). Understanding this formula is critical for solving quadratic equations efficiently.

Geometric Series Sum Formula

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ratio. An infinite geometric series is one that continues indefinitely. The sum of an infinite geometric series can be calculated using the formula:\[ S = \frac{a}{1-r}, \]where \(a\) is the first term and \(r\) is the common ratio, provided that \(|r| < 1\). This condition ensures that the terms decrease sufficiently, allowing the series to converge to a finite sum.

In the original exercise, the series is given by \( \sum_{n=2}^{\infty}(1+c)^{-n} \),where the first term \(a = (1+c)^{-2}\) and the common ratio \(r = \frac{1}{1+c}\). Applying the sum formula helps us understand how the series converges and allows us to set up an equation to solve for \(c\). The condition \(|r| < 1\) is crucial because it limits the feasible range of \(c\), ensuring the series adds up to the specified value.

Problem Solving Steps

Problem-solving in mathematics often requires a systematic approach to understand, organize, and solve the problem efficiently. Here are some steps that can be applied:

• Understand the Problem: Begin by fully grasping what is being asked. Identify the type of problem and the key variables involved.
• Write Down Known Information: List all given information and translate the problem into mathematical expressions or equations.
• Apply Relevant Formulas: Use appropriate mathematical formulas to transform and reorganize the information you have.
• Solve Algebraically: Work through the algebraic operations necessary to isolate the variable of interest or variables.
• Verify and Validate: Once a solution is found, check if it makes sense considering the original question and constraints.

In our exercise, we used these steps effectively to identify and apply both the infinite geometric series formula and the quadratic formula, ultimately solving for the variable \(c\). By following these clear problem-solving steps, complex mathematical problems become more manageable, and solutions become easier to find.