Problem 111 - Essential Calculus

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(|r|<1,\) then \(\sum_{n=1}^{\infty} a r^{n}=\frac{a}{(1-r)} .\)

Step-by-Step Solution

Answer

The statement is false because the summation should start from n = 0, not n = 1. This can be demonstrated using a counterexample, such as when \(a = 3\) and \(r = 1/2\), the sum returns an incorrect value if started from n = 1 compared to the correct value when started from n = 0.

Step 1: Verify the result from Geometric Series Summation

The sum of an infinite geometric series \(\sum_{n=0}^{\infty} a r^{n}\) is given by a / (1 - r), provided that the absolute value of 'r' is less than 1. This equation gives a finite sum when |r| < 1.

Step 2: Correct the Provided Statement

The given statement is almost correct, but there is a subtle issue related to the starting index of the sum. The correct summation starts from n = 0 not n = 1. This discrepancy makes the statement false.

Step 3: Provide a Counterexample

For an example, consider \(a = 3\) and \(r = 1/2\). According to the statement, the sum would start from n = 1, hence be \(3/2 + 3/4 + 3/8 + ...\). This isn't equal to \(3 / (1 - (1/2)) = 6\), while if we start from n = 0 giving us \(3 + 3/2 + 3/4 + ...\) which equals 6. Therefore, the statement is false.

Key Concepts

Infinite Series

An infinite series is a sum of infinitely many terms that follows a specific pattern. In the context of a geometric series, each term is obtained by multiplying the previous term by a constant, known as the common ratio. Geometric series are a key example of infinite series, marked by their predictable progression of terms, such as:

• The first term is denoted by 'a', which is the starting value of the series.
• Each successive term is the product of the previous term and the constant ratio 'r'.

For example, the series \(a, ar, ar^2, ar^3, \ldots \) showcases the simplest form of geometric progression that can extend to infinity. An understanding of infinite series like this is crucial in many areas of mathematics, particularly in problems involving summation and convergence.

Convergence

Convergence is a crucial concept when dealing with infinite series. It tells us whether the sum of all the terms in the series approaches a specific number as more and more terms are added in. In the realm of geometric series, convergence happens if the absolute value of the common ratio 'r' is less than 1. When this condition is met, the infinite series sum, \(\sum_{n=0}^{\infty} ar^{n}\), converges to a finite value. This finite value can be found using the formula:

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

Convergence ensures that an infinite series has a meaningful total. Without convergence, the series doesn't sum up to a finite number and can be considered divergent. Recognizing convergence is essential for validating mathematical claims like the ones encountered in theoretical proofs and applied mathematics.

Mathematical Proof

Mathematical proof is a method of demonstrating the truth of a statement through logical reasoning and established principles. In proving properties of series, like those of geometric series, we outline why certain equations and statements hold true or, as seen in the original exercise, why they might be false.In this specific case, the concept of proof comes into play to correct the original statement. By showing where the statement fails, using a counterexample, the proof clarifies that for the geometric series formula \(\frac{a}{1-r}\) to be applied correctly, the right starting index should be used. Here, the proof entails:

• Identifying the terms of the series as starting either from 'n=0' or 'n=1'.
• Calculating the series sum based on the chosen start point.
• By demonstrating the discrepancy, we utilize the proof to highlight errors in assumptions and align conceptual understanding with accurate application.

Proofs are fundamental in mathematics for establishing the validity of concepts and ensuring they align with logical standards.