Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove the following formula for a geometric sum: $$ \sum_{k=0}^{n} a r^{k}=a+a r+a r^{2}+\cdots+a r^{n}=\frac{a-a r^{n+1}}{1-r}(r \neq 1) $$

Step-by-Step Solution

Answer

The sum is \( \frac{a - ar^{n+1}}{1 - r} \) for \( r \neq 1 \).

Step 1: Consider the Sum

Write the given sum in expanded form as \( S = a + ar + ar^2 + \dots + ar^n \). We're tasked with proving this can be expressed with the formula \( \frac{a - ar^{n+1}}{1 - r} \) for \( r eq 1 \).

Step 2: Multiply by r

Multiply the entire sum by \( r \) resulting in \( rS = ar + ar^2 + \cdots + ar^{n+1} \). This aligns each term with the subsequent term in the original series.

Step 3: Subtract the Two Equations

Subtract \( rS \) from \( S \) to eliminate most terms in the series: \( S - rS = (a + ar + ar^2 + \cdots + ar^n) - (ar + ar^2 + \cdots + ar^{n+1}) \). Observe that this leaves: \( S - rS = a - ar^{n+1} \).

Step 4: Factor and Solve for S

Factor the left side: \( S(1 - r) = a - ar^{n+1} \). Solve for \( S \) by dividing by \( 1 - r \): \( S = \frac{a - ar^{n+1}}{1 - r} \).

Step 5: Conclusion

We've shown through manipulation of the geometric sum that the closed-form formula for the sum is correctly expressed as given by \( \sum_{k=0}^{n} a r^{k} = \frac{a - ar^{n+1}}{1 - r} \), provided \( r eq 1 \).

Key Concepts

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. The expression for a geometric progression is typically written as:

• First term ( \(a\) )
• Common ratio ( \(r\) )

This kind of sequence appears often in various mathematical contexts and is fundamental in understanding geometric series.
For instance, in the formula given by the exercise, \(\sum_{k=0}^{n} a r^{k}\), the sequence \(a, ar, ar^2, \dots, ar^n\) forms a geometric progression. Each term is found by multiplying \(ar^{k-1}\) by \(r\) to get \(ar^{k}\). Geometric progressions are vital because they're used to model phenomena with exponential growth or decay, such as population growth or radioactive decay.

Mathematical Proof

The mathematical proof of a formula usually involves logical reasoning and often includes the manipulation and transformation of expressions to arrive at a desired equation. For the problem at hand, we need to prove the formula: \[ S = \frac{a - ar^{n+1}}{1 - r} \] Here, our task is to show this formula indeed represents the sum of a geometric progression when \(r eq 1\).To tackle this:

• First, represent the sum of the sequence. This forms our initial equation.
• Then, identify transformations (like multiplying by \(r\) and aligning terms) that aid in simplifying the expression.
• Finally, simplify and rearrange until reaching the target formula.

The steps illustrated show how one can methodically transform the series to verify its closed-form equation is valid.

Series Convergence

Convergence in a series refers to the behavior of the sum as more terms are added. A geometric series can converge or diverge, depending upon the common ratio \(r\).- For \(|r| < 1\), the series tends to converge. Here, as \(n\) increases, \(ar^{n+1}\) approaches zero, making the formula \(\frac{a}{1 - r}\) appropriate for the sum because the terms decrease rapidly. - For \(|r| \geq 1\), the series is typically divergent because the terms remain large or increase, leading to an undefined or infinite sum.Convergence is crucial in applications where precise values are necessary—like in financial modeling or physics—for ensuring calculations remain accurate as more terms continue the patterns dictated by resemblance in ratios. Understanding convergence helps anticipate the long-term behavior of a sequence's sum.

Algebraic Manipulation

Algebraic manipulation involves using algebraic rules to simplify or reorganize expressions. It is a fundamental skill used extensively in deriving sums or solving equations, such as the given geometric series problem.In solving the geometric sum formula, several manipulations occur:

• Multiplication of the entire series by \(r\) to shift and align terms for subtraction.
• Subtracting aligned series equations allows elimination of most series terms, simplifying understanding of the expression.
• Factoring expressions to isolate \(S\) (the sum) and solve in terms of variables \(a\) and \(r\).

These steps show how strategic choices in manipulation can effectively prove mathematical statements and expose the intrinsic value of expression components.