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Chapter 1: Problem 186

For a real number \(x>1\), we repeatedly replace \(x\) by \(x-\sqrt[2011]{x}\) until the result is at most 1. Let \(N(x)\) be the number of replacement steps that is needed. Determine, with proof, whether the improper integral \(\int_1^{\infty} \frac{N(x)}{x^2} d x\) converges.

Short Answer

Expert verified
The improper integral converges.

Step by step solution

01

Understand the Given Function and Replacement

For a real number \(x > 1\), the replacement defined is \(x \rightarrow x - \root{2011}{x}\).
02

Analyze the Number of Replacement Steps

The function \(N(x)\) represents the number of times this replacement must be performed until \(x \rightarrow 1\).
03

Establish the Behavior of the Function

Observe how the value of \(x\) changes with each replacement step. Since \(x\) is always replaced by \(x - x^{1/2011}\), the change in \(x\) is very small for large \(x\).
04

Approximate the Behavior for Large \(x\)

For large \(x\), approximating the number of steps can be crucial. We approximate the change \(\frac{dx}{dn} \thickapprox -x^{1/2011}\), leading to \(\frac{dn}{dx} \thickapprox -x^{-1/2011}\).
05

Integrate the Steps Needed

Integrate to find the number of steps: \(N(x) \thickapprox \frac{2011}{2010} x^{2010/2011}\).
06

Set Up the Improper Integral

Evaluate the improper integral: $$\begin{aligned}\textcolor{Green}\boxed{\textcolor{black}{I = \textcolor{DeepPink}{\frac{2011}{2010}}}} \textcolor{KuwaitBlue}{\begin{cases} \textcolor{Red}{\textcolor{Red}{\frac{2011}{2010}}} \textcolor{Emerald}{\frac{1}{x^{1/2011}}} \textcolor{Blue}{N(x) \thickapprox \textcolor{Orange}{\frac{1}{x^2}} dx\right.}}.$$
07

Evaluate the Integral's Convergence

Assess convergence by evaluating the integral boundaries from \(1\) to \(\infty\). Observe that the exponent on \(x\) results in convergence or divergence.
08

Proof of Convergence

Since \(N(x) \thickapprox x^{2010/2011}\), the integral becomes: $$I = \textcolor{Red}{\frac{2011}{2010}} \textcolor{Red}{\frac{\begin{tiny}} \frac{\textcolor{Gray}{2010}} dt {}\frac{x^{2010/2011}}Consider \textcolor{Black}{\textcolor{lavenderblush}{\textcolor{Black}{\textcolor{codeforest}{\textwithcolor{functionsteps:steps:}}}}}}.

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Key Concepts

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

improper integral
An improper integral involves integrating a function over an unbounded interval or integrating a function that becomes unbounded within the interval. In our context, we look at the improper integral \(\int_1^{\infty} \frac{N(x)}{x^2} dx\). Since the upper limit is infinity, it is an improper integral. Integrating functions from 1 to infinity can be tricky, as we have to determine if the function 'behaves' well enough to produce a finite result.

The main steps involve:
  • Breaking down the function and understanding its behavior over large scales of x.
  • Performing a detailed analysis of N(x), which tells how many replacement steps take place as x decreases.
  • Setting up the integral and using known mathematical techniques to ensure convergence or divergence.
Thus, the proper understanding of function behavior is critical to solving improper integrals.
sequence convergence
Sequence convergence occurs when the terms of a sequence approach a specific value as the index increases to infinity. In our problem, as x tends toward infinity, each replacement operation gradually diminishes x based on the function \(x - x^{1/2011}\).

For sequence convergence:
  • We analyze how x changes with each replacement. For large values of x, the change x - x^{1/2011} is tiny.
  • We track this decrement over numerous steps. By approximating how x evolves step by step, we can visualize and measure the total number of steps required until x reduces to 1.
  • Understanding the decrement behavior is key. When approximating \(\frac{dx}{dn} \thickapprox -x^{1/2011}\), and transforming it to \(\frac{dn}{dx} \thickapprox -x^{-1/2011}\).
This approach helps in estimating N(x) accurately and validates it against convergence criteria.
question solving steps
Solving this problem step by step ensures clarity and correctness. Follow these simplified steps:
  • Step 1: Understand the problem and the replacement function given by \(x \rightarrow x - \sqrt[2011]{x}\).
  • Step 2: Define N(x), the number of replacement steps till x reaches <= 1.
  • Step 3: Observe how x changes with each operation and understand decrement nature.
  • Step 4: Approximate \(\frac{dx}{dn} \thickapprox -x^{1/2011}\) and eventually \(N(x) \thickapprox \frac{2011}{2010} x^{2010/2011}\).
  • Step 5: Set up the improper integral \(\int_1^{\infty} \frac{N(x)}{x^2} dx\) based on this approximation.
  • Step 6: Evaluate the integral to determine convergence (by comparing powers of x in the integral).
  • Step 7: Conclude based on the resulting integral bounds and behavior.
By following these steps thoroughly, one can judge whether the improper integral converges or diverges. In this scenario, we observe that the function inside the integral \( \frac{N(x)}{x^2}\) is well-behaved and leads to an overall conclusion about its convergence.

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