Advanced Search
All Posts : Applications, Examples and Libraries
 

At a recent undegraduate competition the students had to compute the following limit

 

> 

Limit( n * Diff( (exp(x)-1)/x, x$n), n=infinity ) assuming x<>0;

Limit(n*(Diff((exp(x)-1)/x, `$`(x, n))), n = infinity)

 (1)

 

Maple is able to compute the symbolic n-fold derivative and I hoped that the limit will be computed at once.

Unfortunately it is not so easy.
Maybe someone finds a more more straightforward way.

 

> 

restart;
> 

f := n * diff( (exp(x)-1)/x, x$n );

n*(-1/x)^n*(-GAMMA(n+1)+GAMMA(n+1, -x))/x

 (2)
> 

limit(%, n=infinity);

limit(n*(-1/x)^n*(-GAMMA(n+1)+GAMMA(n+1, -x))/x, n = infinity)

 (3)
> 

simplify(%) assuming x>0;

limit(-(-1)^n*x^(-n-1)*n*(GAMMA(n+1)-GAMMA(n+1, -x)), n = infinity)

 (4)

 

So, Maple cannot compute directly the limit.

 

> 

convert(f, Int) assuming n::posint;

-n*(-1/x)^n*(-x)^(n+1)*GAMMA(2+n)*(Int(exp(_t1*x)*_t1^n, _t1 = 0 .. 1))/((n+1)*(Int(_k1^n*exp(-_k1), _k1 = 0 .. infinity))*x)

 (5)
> 

J:=simplify(%)  assuming n::posint;

n*(Int(exp(x*_k1)*_k1^n, _k1 = 0 .. 1))*GAMMA(n+1)/(Int(_k1^n*exp(-_k1), _k1 = 0 .. infinity))

 (6)
> 

L:=convert(J, Int) assuming n::posint;

n*(Int(exp(x*_k1)*_k1^n, _k1 = 0 .. 1))

 (7)
> 

L:=subs(_k1=u, L);

n*(Int(exp(x*u)*u^n, u = 0 .. 1))

 (8)

 

Now it should be easy, but Maple needs help.

 

> 

with(IntegrationTools):
> 

L1:=Change(L, u^n = t, t) assuming n::posint;

Int(exp((x*t^(1/n)*n+ln(t))/n), t = 0 .. 1)

 (9)
> 

limit(L1, n=infinity);  # OK

exp(x)

 (10)
> 

####################################################################

Note that the limit can also be computed using an integration by parts, but Maple refuses to finalize:

> 

Parts(L, exp(u*x)) assuming n::posint;

n*(exp(x)/(n+1)-(Int(u^(n+1)*x*exp(x*u)/(n+1), u = 0 .. 1)))

 (11)
> 

simplify(%);

n*(-x*(Int(u^(n+1)*exp(x*u), u = 0 .. 1))+exp(x))/(n+1)

 (12)
> 

limit(%, n=infinity);

limit(n*(-x*(Int(u^(n+1)*exp(x*u), u = 0 .. 1))+exp(x))/(n+1), n = infinity)

 (13)
> 

value(%);  # we are almost back!

limit(n*((-x)^(-n)*(-(n+1)*n*GAMMA(n)/x-(-x)^n*(x-n-1)*exp(x)/x+(n+1)*n*GAMMA(n, -x)/x)+exp(x))/(n+1), n = infinity)

 (14)
> 

 

limit int

Please Wait...