Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems , determine whether the series converges or diverges. Explain your reasoning. \(\frac{1}{2 \cdot 2^{2}}+\frac{1}{3 \cdot 2^{3}}+\frac{1}{4 \cdot 2^{4}}+\frac{1}{5 \cdot 2^{5}}+\cdots\) (Hint: Compare this term-by-term to a geometric series you know. Choose a convergent geometric series whose terms are larger than the terms of this series.)

Step-by-Step Solution

Answer

The series \(\frac{1}{2*2^{2}}+\frac{1}{3*2^{3}}+\frac{1}{4*2^{4}}+\frac{1}{5*2^{5}}+\cdots\) converges according to the comparison test with a convergent geometric series.

Step 1: Simplify the series

We first simplify the series by separating the two terms in the denominator, which gives: \(\frac{1}{2*2^{2}}+\frac{1}{3*2^{3}}+\frac{1}{4*2^{4}}+\frac{1}{5*2^{5}}+\cdots\) = \(\frac{1}{2^{3}}+\frac{1}{2^{4}}+\frac{1}{2^{5}}+\frac{1}{2^{6}}+\cdots\))

Step 2: Identify the related geometric series

The series \(\frac{1}{2^{3}}+\frac{1}{2^{4}}+\frac{1}{2^{5}}+\frac{1}{2^{6}}+...\) we have is a geometric series with first term \(\frac{1}{2^{3}}\) and common ratio \(\frac{1}{2}\). A geometric series converges if the absolute value of the common ratio is less than 1, and diverges otherwise.

Step 3: Apply the comparison test

In order to apply the comparison test, we need to compare our series with the geometric series identified. The terms of our series are \(\frac{1}{2^{n+1}}\) and terms of the geometric series are \(\frac{1}{2^n}\). Clearly all the terms of our series are less than the corresponding terms of the geometric series. Since the geometric series is convergent (because \(\frac{1}{2}\) is less than 1), and every term of our series is less than the corresponding term of the convergent geometric series, by the comparison test, our series also converges.

Key Concepts

Comparison Test

The comparison test is a powerful tool when determining the convergence of an infinite series.
It involves comparing a series to a second, known series to draw conclusions about the behavior of the first.
This method relies on the sequence terms of one series being larger or smaller than another known series. To use the comparison test:

• Identify a reference series that is simpler or well-known.
• Check that all terms of the series in question are smaller than the corresponding terms of the reference if the reference converges, or larger if the reference diverges.
• If the known reference series converges and the question series' terms are lesser, then the question series converges by comparison.
• Alternatively, if the reference series diverges and the question series' terms are larger, then the question series diverges.

In the original exercise, the series is compared with a geometric series.
Since all terms of the given series are less than those of the known convergent geometric series, the given series converges by the comparison test.

Geometric Series

A geometric series is one of the simplest types of series in mathematics characterized by a constant ratio between successive terms. A geometric series can be expressed in the form:

• The first term: \(a\)
• Common ratio: \(r\)
• General form: \(a + ar + ar^2 + ar^3 + \ldots\)

The convergence of a geometric series depends largely on the value of the common ratio \(r\):

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

In the provided exercise, the series after simplification was found to be geometric with a first term of \(\frac{1}{2^3}\) and a common ratio of \(\frac{1}{2}\).
Since the absolute value of the common ratio is less than 1, it confirms the series converges.

Mathematical Reasoning

Mathematical reasoning allows us to deduce results based on established principles and logical steps.
It involves understanding and applying mathematical concepts to rationalize solutions. The given problem required using reasoning to decide series convergence.
There are several key steps involved:

• Identify the nature of the series: Recognize any patterns or simplified forms.
• Apply the right tests: Select suitable convergence tests, like the comparison test.
• Draw conclusions: Based on the test results, deduce whether the series converges or diverges.

In the solution, reasoning was used to break down the series and apply the comparison with a known geometric pattern.
By methodically deducing and verifying through logic and evidence, we ensure our conclusion about the series' behavior is accurate.