Converting a Sum to a Riemann Sum and Finding its Limit

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In summary, the conversation revolved around finding the limit of a series using Riemann sums and converting it to an integral form. The individual was unsure of how to manipulate the ln function in order to put it in the form of (b-a)/n Ʃ (a+(b-a)k/n) and asked for tips on finding the correct function f.
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Sean1218
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Homework Statement



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

Homework Equations


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?
 
  • #3
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?
 

1. What is a Riemann sum?

A Riemann sum is a method used in calculus to approximate the area under a curve by dividing it into smaller rectangles. It is an important concept in understanding integrals and finding their values.

2. How is a Riemann sum calculated?

A Riemann sum is calculated by dividing the area under a curve into smaller rectangles, finding the area of each rectangle, and then adding all of those areas together. The smaller the rectangles, the more accurate the approximation will be.

3. What is the purpose of using a Riemann sum?

The purpose of a Riemann sum is to approximate the area under a curve, which can be used to find the value of an integral. It is also used to visualize and understand the concept of integrals.

4. What is the difference between a left, right, and midpoint Riemann sum?

A left Riemann sum uses the left endpoint of each rectangle to calculate its area, while a right Riemann sum uses the right endpoint. A midpoint Riemann sum uses the midpoint of each rectangle. These different methods can result in slightly different approximations of the area under a curve.

5. How does a Riemann sum relate to limits?

A Riemann sum is essentially a way of approximating the area under a curve, but as the number of rectangles used approaches infinity, the approximation becomes more accurate and approaches the exact value of the integral. This concept relates to limits, as the limit of the Riemann sum is the exact value of the integral.

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