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Chapter 10: Problem 65

$$\text { Let } a_{n}=\left\\{\begin{array}{ll} n / 2^{n}, & \text { if } n \text { is a prime number } \\ 1 / 2^{n}, & \text { otherwise } \end{array}\right.$$ Does \(\sum a_{n}\) converge? Give reasons for your answer.

Short Answer

Expert verified
The series \(\sum a_n\) converges as both terms, under their conditions, produce convergent series.

Step by step solution

01

Understand the Sequence

First, notice that the sequence \(a_n\) is defined piecewise. For prime \(n\), \(a_n = \frac{n}{2^n}\) and for non-prime \(n\), \(a_n = \frac{1}{2^n}\). We are interested in determining whether the infinite series \(\sum a_n\) converges.
02

Consider the General Behavior

Observe that in both cases, as \(n\) becomes large, the terms \(\frac{n}{2^n}\) and \(\frac{1}{2^n}\) tend towards zero. This is because the denominator \(2^n\) grows exponentially compared to the linear growth of \(n\). However, this is not sufficient alone to determine convergence.
03

Apply the Ratio Test

To decisively check for convergence, use the ratio test. For any \(a_n\), the ratio test examines \(\lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|\). Evaluate this test separately for prime and non-prime terms. Since both sequences individually approach zero and decrease, this mixed sequence satisfies the conditions of the ratio test for convergence criteria.
04

Verify the Series' Convergence

For non-prime \(n\), \(\sum \left(\frac{1}{2^n}\right)\) is a geometric series with common ratio \(\frac{1}{2}\) and converges. For prime \(n\), note that \(\frac{n}{2^n}\) reduces faster than \(\frac{1}{n^p}\) for any \(p > 1\). The series \(\sum\frac{1}{n^p}\) converges, implying \(\sum \frac{n}{2^n}\) converges too.
05

Conclude the Overall Convergence

Combine results from evaluating both components of \(a_n\). Both series, \(\sum \frac{n}{2^n}\) and \(\sum \frac{1}{2^n}\), converge. Therefore, the overall series \(\sum a_n\) converges by term-wise convergence.

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Key Concepts

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

Piecewise Function
A piecewise function is a function composed of different expressions based on the input value. Understanding these helps to tackle problems where the function behavior changes under certain conditions. For the given problem, the series for \(a_n\) is defined in two parts based on whether \(n\) is a prime number or not:
  • For prime \(n\): \(a_n = \frac{n}{2^n}\)
  • For non-prime \(n\): \(a_n = \frac{1}{2^n}\)
Piecewise functions are often used in mathematics because many real-world scenarios have different outcomes under different conditions. This allows us to check convergence of each part individually, which simplifies the study of the entire function.
Ratio Test
The ratio test is a powerful tool to test for the convergence of an infinite series. It examines the limit of the absolute value of the ratio of successive terms. For a series \( \sum a_n \), the test checks the following limit:
\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
If \(L < 1\), the series converges. If \(L > 1\) or \(L\) does not exist, the series diverges. For the given series, the terms \(a_n\) for both prime and non-prime numbers decrease towards zero, suggesting that their ratios also approach a limit that satisfies convergence criteria.
  • This method is beneficial when other tests might be complicated or infeasible.
Geometric Series
A geometric series is a series where each term is a constant multiple of the previous term. It is defined by the first term \(a\) and the common ratio \(r\):
\[ \sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + \cdots \]
A geometric series converges if the absolute value of the common ratio \(|r| < 1\). In the sample problem, the series \( \sum \frac{1}{2^n} \) is a geometric series with a common ratio \(r = \frac{1}{2}\), which is less than one. Thus, it converges.
  • Recognizing a geometric series within a more complex series can significantly simplify the process of checking for convergence.
Prime Numbers
Prime numbers are the integers greater than one with no divisors other than one and themselves. They have significant importance in the field of number theory and are often a basis for determining behaviors in mathematical sequences and series.
  • In our current series, the behavior of \(a_n\) changes based on whether \(n\) is prime or not, showcasing how primes can influence the nature of a sequence.
  • Prime numbers appear irregularly spread among natural numbers, making certain calculations and predictions unique when involving them.
Understanding how primes interact within a piecewise function could help in breaking down the problem into more manageable parts. In our series, prime terms \(\left(\frac{n}{2^n}\right)\) decay rapidly enough to ensure convergence.

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Most popular questions from this chapter

Replace \(x\) by \(-x\) in the Taylor series for \(\ln (1+x)\) to obtain a series for \(\ln (1-x)\). Then subtract this from the Taylor series for \(\ln (1+x)\) to show that for \(|x|<1\) $$\ln \frac{1+x}{1-x}=2\left(x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdots\right)$$

Assume that each sequence converges and find its limit. \(2,2+\frac{1}{2}, 2+\frac{1}{2+\frac{1}{2}}, 2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}, \dots\)

For a sequence \(\left\\{a_{n}\right\\}\) the terms of even index are denoted by \(a_{2 k}\) and the terms of odd index by \(a_{2 k+1} .\) Prove that if \(a_{2 k} \rightarrow L\) and \(a_{2 k+1} \rightarrow L,\) then \(a_{n} \rightarrow L\).

a. Assuming that \(\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0\) if \(c\) is any positive constant, show that $$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{t}}=0$$ if \(c\) is any positive constant. b. Prove that \(\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0\) if \(c\) is any positive constant. (Hint: If \(\epsilon=0.001\) and \(c=0.04,\) how large should \(N\) be to ensure that \(\left.\left|1 / n^{c}-0\right|<\epsilon \text { if } n>N ?\right)\)

A patient takes a 300 mg tablet for the control of high blood pressure every morning at the same time. The concentration of the drug in the patient's system decays exponentially at a constant hourly rate of \(k=0.12\) a. How many milligrams of the drug are in the patient's system just before the second tablet is taken? Just before the third tablet is taken? b. In the long run, after taking the medication for at least six months, what quantity of drug is in the patient's body just before taking the next regularly scheduled morning tablet?
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