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Chapter 18: Problem 4

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.)

Short Answer

Expert verified
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 by step solution

01

Simplify the series

We first simplify the series by separating the two terms in the denominator, which gives: \(\frac{1}{2*2^{2}}+\frac{1}{3*2^{3}}+\frac{1}{4*2^{4}}+\frac{1}{5*2^{5}}+\cdots\) = \(\frac{1}{2^{3}}+\frac{1}{2^{4}}+\frac{1}{2^{5}}+\frac{1}{2^{6}}+\cdots\))
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Brent and Rob were working on their math homework when Rob got a headache. Because Rob was incapacitated, Brent went to take a nap. Due to the headache he is blaming on the homework, Rob takes two aspirin. In the body aspirin metabolizes into salicylic acid, which has a half-life of two to three hours. (Source: The pharmacist at a CVS Pharmacy.) Rob is a big fellow, so for the purposes of this problem we ll say three hours is the half-life of salicylic acid. (a) The math headache is haunting him, so three hours later Rob takes two more aspirin. In fact, the headache is so bad that every three hours he takes two more aspirin. If he keeps this up inde nitely, will the level of salicylic acid in his body ever reach the level equivalent to taking four aspirin all at once? (b) Brent wakes up from a deep sleep, looks at his math homework, gets a headache, looks at Rob, and decides that he s going to take two aspirin every two hours. If he keeps this up inde nitely, will the level of salicylic acid in his body every reach the level equivalent to taking three aspirin all at once? Four aspirin all at once? Five aspirin all at once? (Assume again that the half-life of salicylic acid is three hours.)

Write the sum using summation notation. $$ 2-3+4-5+6-\cdots+100 $$

People who have slow metabolism due to a malfunctioning thyroid can take thyroid medication to alleviate their condition. For example, the boxer Muhammad Ali took Thyrolar 3, which is 3 grains of thyroid medication, every day. The amount of the drug in the bloodstream decays exponentially with time. The half-life of Thyrolar is 1 week. (a) Suppose one 3 -grain pill of Thyrolar is taken. Write an equation for the amount of the drug in the bloodstream \(t\) days after it has been taken. (Hint: In part (a) you are dealing with one 3 -grain pill of Thyrolar. Knowing the half-life of Thyrolar, you are asked to come up with a decay equation. This part of the problem has nothing to do with geometric sums.) (b) Suppose that Ali starts with none of the drug in his bloodstream. If he takes 3 grains of Thyrolar every day for ve days, how much Thyrolar is in his bloodstream immediately after having taken the fth pill? (c) Suppose Ali takes 3 grains of Thyrolar each day for one month. How much thyroid medication will be in his bloodstream right before he takes his 31 st pill? Right after? (d) After taking this medicine for many years, what was the amount of the drug in his body immediately after taking a pill? Historical note: Before one of his last ghts Muhammad Ali decided to up his dosage to 6 grains. In doing so he mimicked the symptoms of an overactive thyroid. The result in terms of the ght was dismal.

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ m q+m^{2} q^{4}+m^{3} q^{7}+\cdots+m^{11} q^{31} $$

Write each of the following series rst as a repeating decimal and then as a fraction. (a) \(2+\frac{2}{10}+\frac{2}{100}+\frac{2}{1000}+\cdots\). (b) \(3+\frac{12}{10^{2}}+\frac{12}{10^{4}}+\frac{12}{10^{6}}+\cdots\)
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