Question

Determine whether the following statements are true and give an explanation or counterexample. a. \(\sum_{k=1}^{\infty}\left(\frac{\pi}{e}\right)^{-k}\) is a convergent geometric series. b. If \(a\) is a real number and \(\sum_{k=12}^{\infty} a^{k}\) converges, then \(\sum_{k=1}^{\infty} a^{k}\) converges. c. If the series \(\sum_{k=1}^{\infty} a^{k}\) converges and \(|a|<|b|\), then the series \(\sum_{k=1}^{\infty} b^{k}\) converges.

Short answer

Question: Determine the truth of the following statements and provide an explanation or counterexample for each. a. The series \(\sum_{k=1}^{\infty} (\frac{1}{\pi/e})^{k}\) converges and is a geometric series. b. If the series \(\sum_{k=12}^{\infty} a^{k}\) converges, then the series \(\sum_{k=1}^{\infty} a^{k}\) also converges for any real number a. c. If the series \(\sum_{k=1}^{\infty} a^{k}\) converges and \(|a| < |b|\), then the series \(\sum_{k=1}^{\infty} b^{k}\) also converges. Answer: a. True. The series is convergent and geometric since a = 1, r = \(\frac{1}{\pi/e}\), and \(0 < |\frac{1}{\pi/e}| < 1\). b. True. The convergence of \(\sum_{k=1}^{\infty} a^{k}\) relies on the convergence of the finite sum \(\sum_{k=1}^{11} a^{k}\), which is always convergent. c. False. A counterexample can be found when a = 1/2 and b = 2, in which the first series converges, but the second does not.

Step-by-step solution

Statement a: Determine Convergence and Geometric Property

Recall that a geometric series is of the form \(\sum_{k=1}^{\infty} ar^{k-1}\), where a is a constant and |r| < 1 for convergence. We can rewrite the given series as \(\sum_{k=1}^{\infty} (\frac{1}{\pi/e})^{k}\). Here, a = 1, and r = \(\frac{1}{\pi/e}\). Since \(0 < |\frac{1}{\pi/e}| < 1\), the series converges and is a geometric series. Therefore, statement a is true.

Statement b: Using Convergence Property for Real Number a

Consider the series: \(\sum_{k=12}^{\infty} a^{k}\), which converges. We need to check if this implies that the series \(\sum_{k=1}^{\infty} a^{k}\) also converges. To do this, note the relationship between these two series: \(\sum_{k=1}^{\infty} a^{k} = \sum_{k=1}^{11} a^{k} + \sum_{k=12}^{\infty} a^{k}\). Since \(\sum_{k=12}^{\infty} a^{k}\) converges, the convergence of \(\sum_{k=1}^{\infty} a^{k}\) relies on the convergence of the sum \(\sum_{k=1}^{11} a^{k}\). However, this sum has only 11 terms, so it is always convergent (it's a finite sum). Thus, statement b is true.

Statement c: Convergence Based on Absolute Values

Let's assume that the series \(\sum_{k=1}^{\infty} a^{k}\) converges, and we need to check if the series \(\sum_{k=1}^{\infty} b^{k}\) also converges with \(|a| < |b|\). Observe that \(\sum_{k=1}^{\infty} a^{k}\) is a geometric series with |a| < 1 (because it's convergent). Since \(|a| < |b|\), it is possible for |b| ≥ 1. For example, let's consider a = 1/2 and b = 2. Then the series \(\sum_{k=1}^{\infty} a^{k}\) converges, but the series \(\sum_{k=1}^{\infty} b^{k}\) doesn't converge, because |b|=2 ≥ 1. Thus, statement c is false, and we found a counterexample.

Key concepts

Geometric Series

A geometric series is a sum of the form \( \sum_{k=1}^{\infty} ar^{k-1} \), where \( a \) is a constant and \( r \) is the common ratio. This series will converge, or "settle" to a finite value, if the absolute value of the common ratio \( |r| \) is less than 1. If \( |r| \geq 1 \), the series becomes divergent and will not sum to a finite value.
For example, consider the series \( \sum_{k=1}^{\infty} \left(\frac{1}{3}\right)^k \). Here, \( a = 1 \) and \( r = \frac{1}{3} \), which is clearly less than 1. Thus, the series converges.
Geometric series appear frequently in mathematics due to their simple and regular structure, and their convergence properties are easy to analyze using the ratio \( |r| < 1 \).

Convergence Tests

To determine whether an infinite series converges or diverges, mathematicians use various convergence tests. For geometric series, as noted, the primary test is checking the absolute value of the common ratio. However, for more complex infinite series, other tests are necessary.

• **Ratio Test**: This test involves calculating the limit \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If this limit is less than 1, the series converges.
• **Root Test**: Involves finding \( \lim_{k \to \infty} \sqrt[k]{|a_k|} \). If this is less than 1, the series converges.
• **Integral Test**: For a function \( f(x) \) that is positive, continuous, and decreasing, if the integral from 1 to infinity of \( f(x) \) converges, so does the series \( \sum_{k=1}^{\infty} a_k \).

Understanding these tests helps in analyzing more general forms of series that are not strictly geometric.

Infinite Series

An infinite series is a sum of infinite terms, typically in the form \( \sum_{k=1}^{\infty} a_k \). Such series often challenge our intuition because they do not stop; they go on forever. Despite having infinitely many terms, some infinite series do converge to a finite value.
Infinite series are categorized based on convergence:
- **Convergent Series**: The sum approaches a specific number.
- **Divergent Series**: The sum grows without bound or behaves chaotically.
Analyzing infinite series requires careful application of convergence tests or known rules, such as those applicable to geometric series. It's crucial to understand the behavior of the terms in the series, as these dictate whether the series will "behave" nicely (converge) or continue growing unpredictably (diverge).

Counterexamples in Mathematics

Counterexamples play a vital role in mathematics by proving that a certain statement is false. They are specific examples that contradict a general claim, showing that while a mathematical proposition may seem plausible, it’s not universally true.
Take, for instance, statement c in the original exercise involving the series \( \sum_{k=1}^{\infty} a^{k} \) with \( |a| < |b| \). The series \( \sum_{k=1}^{\infty} a^{k} \) converges, but it was shown with \( a = 1/2 \) and \( b = 2 \) that \( \sum_{k=1}^{\infty} b^{k} \) does not converge because \( b = 2 \) leads to a common ratio \( |b| \geq 1 \).
Using counterexamples helps refine mathematical theories and ensures only the most robust conjectures become accepted truths.