Riemann sum : sum_riemann

sum_riemann takes two arguments : an expression depending of two variables and the list of the name of these two variables.
sum_riemann(expression(n,k),[n,k]) returns in the neighboorhoud of n = + $\infty$ an equivalent of $\sum_{{k=1}}^{n}$expression(n, k) (or of $\sum_{{k=0}}^{{n-1}}$expression(n, k) or of $\sum_{{k=1}}^{{n-1}}$expression(n, k)) when the sum is looked as a Riemann sum associated to a continue function defined on [0,1] or returns "it is probably not a Riemann sum" when the resarch is unavailing.
Exercise 1
Suppose S_n = $$\sum_{{k=1}}^{n}$$$${\frac{{k^2}}{{n^3}}}$$.
Compute $$\lim_{{n \rightarrow +\infty}}^{}$$S_n.
Input :

sum_riemann(k^2/n^3,[n,k])

Output :

1/3

Exercise 2
Suppose S_n = $$\sum_{{k=1}}^{n}$$$${\frac{{k^3}}{{n^4}}}$$.
Compute $$\lim_{{n \rightarrow +\infty}}^{}$$S_n.
Input :

sum_riemann(k^3/n^4,[n,k])

Output :

1/4

Exercise 3
Compute $$\lim_{{n \rightarrow +\infty}}^{}$$($${\frac{{1}}{{n+1}}}$$ + $${\frac{{1}}{{n+2}}}$$ + ... + $${\frac{{1}}{{n+n}}}$$).
Input :

sum_riemann(1/(n+k),[n,k])

Output :

log(2)

Exercise 4
Suppose S_n = $$\sum_{{k=1}}^{n}$$$${\frac{{32n^3}}{{16n^4-k^4}}}$$.
Compute $$\lim_{{n \rightarrow +\infty}}^{}$$S_n.
Input :

sum_riemann(32*n^3/(16*n^4-k^4),[n,k])

Output :

2*atan(1/2)+log(3)