Limit of Sequence: Using Riemann Sum

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Homework Statement


consider the sequence {an} where an= 1/(n+1) + ...+ 1/2n. find its limit.


Homework Equations


the hint given is using riemann sum.


The Attempt at a Solution


we know that since it is increasing and bounded above by one, the sequence converges. I'm not sure where to go from there, especially to use the riemann sum.
 
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Yes this question basically asks you if you know how Riemann or Darboux sums work. Hint: consider the function f defined by f(x) = 1/(1+x) on [0,1]. Choose the most standard partition of an interval you can think of and find out how this relates to your sum.
 
I'm thinking that the partition would be P= 0=x0<x1<x2<...,xn=1 with 1/n partitions. Not sure about how this would relate to the sum that I have in the problem?
 
Yes, that is the right partition if you meant that there are n subintervals each of length 1/n. Now rewrite the original sum as

\sum_{k = 1}^{n}\frac{1}{n+k} = \sum_{k = 1}^{n}\left(\frac{1}{1+\frac{k}{n}}\right)\cdot\frac{1}{n}.

Finally, consider the upper and lower sums.
 
Here is what I've come up with:
let f(x)=x/(x+1)
let P be the partition {x0, x1, ...xn} where xi=i/n and ti=xi and the mesh is 1/n=xi-xi-1 (i and i-1 are subscripts)

then S(P,f)=SUM: f(ti)(xi-xi-1) = SUM: 1/(n+1) * 1/n, from i=1 to n which is the sequence an.
since S(p,f) converges to the integral of 1/(x+1)dx, from 0 to 1, an converges to this as well. so the limit of an is the value of the integral, which is ln2.

Is that where you were going with it?
 
Last edited:
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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