limit sought

[AlfredFlaßhaar]

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...)

[Werner_E]

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:

[Werner_E]

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 ...

[Werner_E · Accepted solution]

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 😉

[AlfredFlaßhaar]

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!

[Cornel]

Post delated by author

[mvenich]

This is what I got in Maple Fl;ow:

[Werner_E]

Great!

Can Maple Flow also handle the original expression (in x) before doing the substitution x^n = y ?

[mvenich]

Looks like it can't

[Werner_E]

Disappointing!

You missed the factor n in front of the integral and so I think the limit simply should be zero.

Adding the factor n we get an expression of the form "infinity * zero" with the limit given above.

But Maple Flow is not on its own.

Mathcad (15) with muPad can't come up with a solution either

and Prime 10 with friCAS/Axiom returns an interesting creative solution. Only the second symbolic evaluation is able to simplify the simple integrals in the first result.

The interesting part is that the correct result "0" is amongst the choices as well as the result for the original task with the additional factor "n".

Not absolutely sure what %Z, %Y should denote - guess it means "x" and its a bug that these internal variables shows up in the final result.

[AlfredFlaßhaar]

A theoretical background for this solution must be mentioned here. The solution requires exchanging the limit and the integral. This is permissible for uniformly continuous functions (here the integrand). Fortunately, according to Heine's theorem, this is fulfilled here and according to the so-called commutation theorem, the interchanges are permitted. Software does not check this.

[Werner_E]

Thanks for the clarification.

Swapping integration and limiting was exactly the point where I was unsure whether and under what conditions it was permissible.

[LucMeekes]

I tried the original problem in Mathcad 11 as described by Alfred, but it failed.

This one doesn't:

Luc