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Recently, a local Math olympiad included the following problem:

 

 

 Compute:Limit(n*(Int(x^n/(x^n+x+2024), x = 0 .. 1)), n = infinity).

 

 

I was almost sure that Maple can compute it, but unfortunately it needed help!

restart;

J:=n->n*int(x^n/(x^n+x+2024),x=0..1);

proc (n) options operator, arrow; n*(int(x^n/(x^n+x+2024), x = 0 .. 1)) end proc

(1)

limit(J(n), n=infinity); # Maple gives up.

limit(n*(int(x^n/(x^n+x+2024), x = 0 .. 1)), n = infinity)

(2)

simplify(IntegrationTools:-Change(J(n), x=t^(1/n), t)) assuming n>1;

int(t^(1/n)/(t+t^(1/n)+2024), t = 0 .. 1)

(3)

ans:=limit(%, n=infinity);

ln(2)+ln(1013)-4*ln(3)-2*ln(5)

(4)

#map(combine,ans, ln);

combine(ans,ln, integer);

4*ln((1/15)*2026^(1/4)*5^(1/2))

(5)

ln(exp(%)); # should be obtainable with convert

ln(2026/2025)

(6)

evalf(% = J(100)); # sanity check;

0.4937051076e-3 = 0.488824e-3+0.*I

(7)


Download lim-int-vv.mw

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Try it yourself, you will understand what I mean. (MF2024.2.1)

 



How to fix these warnings

Maple 2021 asked by JAMET 425 February 08