What is the solution to this infinite series problem?

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Discussion Overview

The discussion revolves around finding the positive integer k for which the infinite series \(\sum \limits_{n=4}^k \frac{1}{\sqrt{n} + \sqrt{n+1}} = 10\). The scope includes mathematical reasoning and exploration of computational tools for solving the problem.

Discussion Character

  • Exploratory, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant asks for a solution without using a calculator, presenting the series problem.
  • Another participant suggests a hint involving multiplying the numerator and denominator by \(\sqrt{n} - \sqrt{n+1}\).
  • A different participant suspects that the series may become a telescoping series.
  • One participant shares their experience using Maxima as a computer algebra system (CAS), noting that it produced a large sum of radicals and a float approximation close to 10.
  • Another participant mentions that rationalizing the denominator first in Maxima leads to the correct answer and inquires about experiences with other CAS tools.
  • One participant claims that the problem simplifies to \(\sum \limits_{n=4}^k - \sqrt{n} + \sqrt{n+1} = 10\), implying a different interpretation of the series.

Areas of Agreement / Disagreement

Participants express differing views on the approach to solving the series, with no consensus on the best method or interpretation of the problem. The discussion remains unresolved regarding the exact solution and the effectiveness of various computational tools.

Contextual Notes

Participants mention limitations in computational tools, such as the output of large sums of radicals and the accuracy of floating-point results. There is also a lack of clarity on the assumptions behind the simplifications proposed.

camilus
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can anyone find a solution without using a calculator??

This is the problem:


Find the positive interger k for which [tex]\sum \limits_{n=4}^k {1 \over \sqrt{n} + \sqrt{n+1}} = 10[/tex]
 
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Hint: Multiply numerator and denominator by sqrt(n) - sqrt(n+1)
 
I suspect this will become something call telescoping series
 
Maxima

Has anyone used Maxima as a CAS? I used Maxima to check my answer to this problem and it just gives me a huge sum of radicals. When I used its float command it gave me 9.9999999999996 as the sum for my answer. Are all of the CAS's this limited?
 
Rationalize for the computer

sennyk said:
Has anyone used Maxima as a CAS? I used Maxima to check my answer to this problem and it just gives me a huge sum of radicals. When I used its float command it gave me 9.9999999999996 as the sum for my answer. Are all of the CAS's this limited?

If I rationalize the denominator first, it then gives me the correct answer. If anyone has any experiences with other CAS's, please share.
 
This is a very easy problem because it reduces to:

[tex]\sum \limits_{n=4}^k - \sqrt{n} + \sqrt{n+1}} = 10[/tex]
 

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