Limit of Sequence: Using Riemann Sum

In summary, the conversation discusses finding the limit of the sequence {an}, where an = 1/(n+1) + ... + 1/2n. The hint given is to use Riemann sum and consider the function f(x) = 1/(1+x) on [0,1]. The suggested partition is P = {x0, x1, ... xn} where xi = i/n and ti = xi, with a mesh of 1/n = xi - xi-1. Finally, the upper and lower sums are considered, leading to the conclusion that the limit of an is ln2.
  • #1
llursweetiell
11
0

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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  • #2
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.
 
  • #3
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?
 
  • #4
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

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

Finally, consider the upper and lower sums.
 
  • #5
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:

Related to Limit of Sequence: Using Riemann Sum

1. What is a limit of sequence?

A limit of sequence is a mathematical concept that describes the behavior of a sequence of numbers as it approaches a specific value or point. It is used to determine the behavior of a sequence as the number of terms in the sequence increases.

2. How is a limit of sequence calculated using Riemann Sum?

Riemann Sum is a method used to approximate the area under a curve. To calculate the limit of sequence using Riemann Sum, the curve is divided into smaller intervals and the sum of the areas of these intervals is taken. As the intervals become smaller, the value of the Riemann Sum approaches the limit of the sequence.

3. What is the significance of the limit of sequence in mathematics?

The limit of sequence is important in mathematics as it helps determine the convergence or divergence of a sequence. It also plays a crucial role in the study of functions, differentiation, and integration.

4. Can the limit of sequence be undefined?

Yes, the limit of sequence can be undefined if the sequence does not approach a specific value or point. This can happen if the terms in the sequence oscillate or if the sequence has no pattern.

5. How can the limit of sequence be used in real-world applications?

The limit of sequence has various applications in real-world problems, such as in physics, engineering, and economics. It can be used to model and analyze continuous processes, such as motion, growth, and decay, and to make predictions about the behavior of these processes.

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