limit sought

Forum|Forum|1 year ago
February 14, 2025
3 replies
1940 views

I recently received the following task from a competition:
Calculate lim(n-->oo) [n*integral(from 0; to 1) (x^n/(x^n+x+2024)].

Using derive, MC15 and Maple I got three different solutions. Can Mathcad solve this task?
(Solution: 0.0004937...)

Calculus_Derivatives

Best answer by Werner_E

The equivalent function g(n), which is obtained by substituting y=x^n, is numerically much more stable!

Nonetheless, Symbolics is still unable to find an exact symbolic solution when applying the limit

However, it is possible to do it this way

which of course is equivalent to

a basic integral for which we would not need any software 😉

W

Werner_E

25-Diamond I

I don't think that Mathcad or Prime are able to arrive at a solution.

The symbolics of both is not able to come up with a solution and the numerics runs into severe round-off errors for higher values of n.

Here the "results" of Mathcad 15 and Prime 10:

W

Werner_E

25-Diamond I

Additional remark:

We can improve numerical accuracy significantly by decreasing the value of TOL.
But there will always be a value for n above which numerical inaccuracies will strike and falsely pull the result towards zero.

In case of TOL=10^-10 its the values above n=9449 which yield inaccurate/wrong results.

We can zoom in to see the results approaching the value you named and then the sudden death of the numerics starting with a specific n-value

If you force the symbolics into numeric float mode by using a decimal like 1.0 instead of just 1, we can get a pretty good approxumation

I had no luck with much larger arguments, though ('endless' calculation).

Also tried Wolfram Alpha, but uncle Wolfram wants us to insert coins to buy more computation time ...

W

Werner_E

Answer

25-Diamond I

The equivalent function g(n), which is obtained by substituting y=x^n, is numerically much more stable!

Nonetheless, Symbolics is still unable to find an exact symbolic solution when applying the limit

However, it is possible to do it this way

which of course is equivalent to

a basic integral for which we would not need any software 😉

A

AlfredFlaßhaar

Author

15-Moonstone

I was already familiar with the last solution for reasons based on theorems of analysis. However, since I have little experience in numerics, I wanted to experience the convergence behavior experimentally, live and in color, so to speak ;-). I have now become somewhat aware of the limitations. Thank you very much!