Convergence of Improper Integral with Limit Evaluation

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In summary, the conversation discusses whether the improper integral \int_{0}^{1}\frac{1}{\sqrt[3]{x}}dx is divergent or convergent. It is determined that it is an improper integral and the limit must be taken as the lower limit approaches 0 to evaluate it. The question also asks if there is a way to determine convergence without using a test, but it is clarified that this is not possible.
  • #1
flyingpig
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Homework Statement



Determine whether the improper integral diverge or converge. If it is convergent, evaluate it with a limit

[tex]\int_{0}^{1}\frac{1}{\sqrt[3]{x}}dx[/tex]Just from inspection, I thought this was a p-series with 1/3 < 1, but then I noticed the limits of integration is from 0 to 1.

So my question is, is there a way, just from inspection, to notice this is convergent without resorting to any test?
 
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  • #2
Just fixed question
 
  • #3
It's not a p-series. It's not any kind of series at all. It's an improper integral. Integrate it and take the limit as the lower limit approaches 0.
 
  • #4
Dick said:
It's not a p-series. It's not any kind of series at all. It's an improper integral. Integrate it and take the limit as the lower limit approaches 0.

I.E. Evaluate:

[tex]\int_{a}^{1}\frac{1}{\sqrt[3]{x}}dx[/tex]

Take the limit of the result as a → 0+.
 

FAQ: Convergence of Improper Integral with Limit Evaluation

Why does this series converge?

The convergence of a series can be determined by examining its terms and their behavior as the number of terms increases. If the terms of the series approach zero or a constant value as the number of terms increases, then the series is said to converge. This means that the sum of the series will approach a finite value as the number of terms increases.

What factors affect the convergence of a series?

The convergence of a series can be affected by several factors, including the behavior of its terms, the order of the terms, and the presence of any patterns or trends within the series. In addition, the convergence of a series may also be influenced by the choice of mathematical operations used to calculate the series.

How can I determine if a series is convergent or divergent?

To determine if a series is convergent or divergent, you can use various tests such as the comparison test, ratio test, or the integral test. These tests involve comparing the given series to a known convergent or divergent series and examining the behavior of the series as the number of terms increases.

Can a series converge to infinity?

No, a series cannot converge to infinity. A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases. If the sum of the terms approaches infinity, then the series is said to be divergent.

How does the convergence of a series relate to its applications in science?

The convergence of a series is important in many scientific applications, particularly in the fields of physics and engineering. Many physical and engineering phenomena can be described using series, and the convergence of these series can provide valuable insights and predictions about these phenomena. Additionally, the convergence of series is also important in numerical analysis and computer science, where it is used to evaluate complex mathematical equations and algorithms.

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