How to convert the limit of a series into an integral?

In summary, the conversation was about converting a limit of a series into an integral. The participants discussed various methods, such as the integral test for convergence and the Euler-Maclaurin formula. The video mentioned in the conversation showed an example of evaluating a summation by converting it into a Riemann sum, which is equal to a definite integral.
  • #1
Adesh
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If I have a limit of a series then how can I convert it into integral. I know to convert a sum into an integral there must be Δx multiplied to each term and this must go zero. Can you please explain me the conversion of limit of series (normal series with no Δx) into an integral.
Thank you.
 
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  • #2
Adesh said:
If I have a limit of a series then how can I convert it into integral. I know to convert a sum into an integral there must be Δx multiplied to each term and this must go zero. Can you please explain me the conversion of limit of series (normal series with no Δx) into an integral.
Thank you.
Are you looking for something like https://en.wikipedia.org/wiki/Integral_test_for_convergence ?
 
  • #4
Adesh said:
I’m looking for solving the series.

Are you asking about "summing the series" - or are you asking about "testing the convergence of the series"?

To test the convergence of a series using the integral test, we do not need to sum it.
 
  • #5
Stephen Tashi said:
Are you asking about "summing the series" - or are you asking about "testing the convergence of the series"?

To test the convergence of a series using the integral test, we do not need to sum it.
I’m asking how to take the limit of a sum of a convergent series. Sir I’m sorry for ambiguity but I’m trying to be clear as much as possible.
 
  • #6
Adesh said:
I’m asking how to take the limit of a sum of a convergent series. Sir I’m sorry for ambiguity but I’m trying to be clear as much as possible.
There is no general formula ##\sum_{\mathbb{N}} f(n) = \int_a^b g(x)\,dx\,.##
 
  • #7
fresh_42 said:
There is no general formula ##\sum_{\mathbb{N}} f(n) = \int_a^b g(x)\,dx\,.##
Can we do this? Where is dx in summation, sir
 
  • #8
Adesh said:
Can we do this? Where is dx in summation, sir
That's the problem, in the sum we have ##dx=1## which is not the same as what the limit ##dx## actually is. Thus the integral criterion and I'm sure there are some error estimations for it, too, is as best as we can get, as far as I know. If there was such a formula, then it certainly would be part of all curricula.
 
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  • #10
Stephen Tashi said:
See if you like the Euler-Mclaurin summation formula https://en.wikipedia.org/wiki/Euler–Maclaurin_formula
I was solving this problem
lim. ∑1/(n+k). {k varies from zero to m). m= nb
n -> ∞

What most people did in my university or even on internet is that they wrote it like this ∫1/(n+x) dx (limit zero to m) m= nb
(b is a constant)
I'm wondering how can you change it into integral as difference between partitions is one and is not changing. I myself used Euler-Maclaurin formula for solving this.
 
  • #11
Adesh said:
I was solving this problem
lim. ∑1/(n+k). {k varies from zero to m). m= nb
n -> ∞

What most people did in my university or even on internet is that they wrote it like this ∫1/(n+x) dx (limit zero to m) m= nb
(b is a constant)
I'm wondering how can you change it into integral as difference between partitions is one and is not changing. I myself used Euler-Maclaurin formula for solving this.
They are not evaluating the summation by writing a related integral. Most likely, they are using the integral to get an upper bound (or possibly upper and lower bounds) on the summation.
 
  • #12
Mark44 said:
They are not evaluating the summation by writing a related integral. Most likely, they are using the integral to get an upper bound (or possibly upper and lower bounds) on the summation.
Sir can you please explain elaborately the conversion. Here is the link where he uses the conversion of series to an integral for solving the problem.
 
  • #13
Adesh said:
If I have a limit of a series then how can I convert it into integral. I know to convert a sum into an integral there must be Δx multiplied to each term and this must go zero. Can you please explain me the conversion of limit of series (normal series with no Δx) into an integral.
The guy in the video is evaluating the summation by converting the summation into the Riemann sum that is equal to a definite integral. See this Wikipedia article: https://en.wikipedia.org/wiki/Riemann_sum

In the video, he starts with ##\lim_{n \to \infty} \sum_{r = 0}^{n - 1} \frac 1 {\sqrt{n^2 - r^2}}##
In the next step, he writes a Riemann sum and the definite integral the Riemann sum is equal to.
##\lim_{n \to \infty} \sum_{r = 0}^{n - 1} f(\frac r n) \frac 1 n = \int_0^1 f(n) dn##
He then works to convert the expression in the summation he's working on, ##\frac 1 {\sqrt{n^2 - r^2}}##, to make it look like ##f(\frac r n)\frac 1 n##. That fraction ##\frac 1 n## is the piece that you were missing, and which confused the people who have responded to your question.That fraction in the summation plays the same role as ##dx## in the integral.
 
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  • #14
Mark44 said:
The guy in the video is evaluating the summation by converting the summation into the Riemann sum that is equal to a definite integral. See this Wikipedia article: https://en.wikipedia.org/wiki/Riemann_sum

In the video, he starts with ##\lim_{n \to \infty} \sum_{r = 0}^{n - 1} \frac 1 {\sqrt{n^2 - r^2}}##
In the next step, he writes a Riemann sum and the definite integral the Riemann sum is equal to.
##\lim_{n \to \infty} \sum_{r = 0}^{n - 1} f(\frac r n) \frac 1 n = \int_0^1 f(n) dn##
He then works to convert the expression in the summation he's working on, ##\frac 1 {\sqrt{n^2 - r^2}}##, to make it look like ##f(\frac r n)\frac 1 n##. That fraction ##\frac 1 n## is the piece that you were missing, and which confused the people who have responded to your question.That fraction in the summation plays the same role as ##dx## in the integral.
Sir, is there any other method to solve it. I shall highly appreciate it.
 
  • #15
Adesh said:
Sir, is there any other method to solve it. I shall highly appreciate it.
I don't believe there is.
 
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  • #16
Mark44 said:
I don't believe there is.
Thank you
 
  • #17
Adesh said:
I was solving this problem
lim. ∑1/(n+k). {k varies from zero to m). m= nb
n -> ∞

What most people did in my university or even on internet is that they wrote it like this ∫1/(n+x) dx (limit zero to m) m= nb
(b is a constant)
I'm wondering how can you change it into integral as difference between partitions is one and is not changing. I myself used Euler-Maclaurin formula for solving this.
Mark44 said:
The guy in the video is evaluating the summation by converting the summation into the Riemann sum that is equal to a definite integral. See this Wikipedia article: https://en.wikipedia.org/wiki/Riemann_sum

In the video, he starts with ##\lim_{n \to \infty} \sum_{r = 0}^{n - 1} \frac 1 {\sqrt{n^2 - r^2}}##
In the next step, he writes a Riemann sum and the definite integral the Riemann sum is equal to.
##\lim_{n \to \infty} \sum_{r = 0}^{n - 1} f(\frac r n) \frac 1 n = \int_0^1 f(n) dn##
He then works to convert the expression in the summation he's working on, ##\frac 1 {\sqrt{n^2 - r^2}}##, to make it look like ##f(\frac r n)\frac 1 n##. That fraction ##\frac 1 n## is the piece that you were missing, and which confused the people who have responded to your question.That fraction in the summation plays the same role as ##dx## in the integral.
With the small caveat that the dx are not required to be of uniform width ( here 1/n) , only that they all have width approaching 0 in the limit.
 
  • #18
Afesh, why don't you try replicating the method , dividing the denominator by n and mikick the rest of the approach in the video described by mark44 and see where that leads you?
 

1. What is the purpose of converting a limit of a series into an integral?

Converting the limit of a series into an integral allows us to find the area under a curve, which can be a more accurate representation of the sum of the series.

2. How do I determine the limits of integration when converting a series into an integral?

The limits of integration can be determined by finding the endpoints of the series, which are often the values of x that the series converges towards.

3. What is the relationship between the limit of a series and the integral of its terms?

The limit of a series is equal to the integral of its terms when the series is convergent.

4. Can the conversion of a series into an integral be used for all types of series?

No, the conversion of a series into an integral is only applicable to series that converge towards a finite value. Divergent series cannot be converted into integrals.

5. Is there a specific method for converting a limit of a series into an integral?

Yes, there are several methods such as the Riemann sum method or the geometric interpretation method that can be used to convert the limit of a series into an integral.

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