Converting a Sum to a Riemann Sum and Finding its Limit

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SUMMARY

The discussion focuses on finding the limit of the expression limn→∞ (1/n)(Ʃk=1n ln(2n/(n+k))). Participants clarify that this expression can be interpreted as a Riemann sum, specifically in the form (1/n)f(k/n) for a function f. The challenge lies in manipulating the logarithmic function to fit the Riemann sum format, which is essential for converting the sum into an integral for limit evaluation.

PREREQUISITES
  • Understanding of Riemann sums and their properties
  • Familiarity with limits in calculus
  • Knowledge of logarithmic functions and their manipulation
  • Basic skills in summation notation and integral calculus
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  • Study the properties of Riemann sums and their applications in calculus
  • Learn techniques for manipulating logarithmic expressions
  • Explore the concept of limits and their evaluation in calculus
  • Investigate the relationship between summation and integration
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Students and educators in calculus, particularly those focusing on limits, Riemann sums, and logarithmic functions. This discussion is beneficial for anyone tackling advanced calculus problems involving summation and integration techniques.

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Homework Statement



Find limn->∞ (1/n)(Ʃk=1 to n ln(2n/(n+k)))

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The Attempt at a Solution



I'm not sure if this is even a riemann sum at all, but I don't see what else it could be. I wanted to find the riemann portion first to get rid of the sigma notation then find the limit of everything, but I don't have any of the information I need to do the riemann sum or to convert it to an integral.
 
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I realize I need to be able to put it in a form of (b-a)/n Ʃ (a+(b-a)k/n), but I'm not sure how I can work with ln and manipulate it when everything is stuck in ln. Any tips?
 
If you can put it in the form of sum (1/n)f(k/n) for some function f then it's a Riemann sum of the function f(x) for x=0 to 1. What's f?
 

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