5.18.3  Riemann sum : sum_riemann

sum_riemann takes two arguments : an expression depending on two variables and the list of the name of these two variables.
sum_riemann(expression(n,k),[n,k]) returns in the neighborhood of n=+∞ an equivalent of ∑_k=1ⁿ expression(n,k) (or of ∑_k=0ⁿ⁻¹ expression(n,k) or of ∑_k=1ⁿ⁻¹ expression(n,k)) when the sum is looked on as a Riemann sum associated to a continuous function defined on [0,1] or returns "it is probably not a Riemann sum" when the no result is found.
Exercise 1
Suppose S_n=∑_k=1ⁿ k²/n³.
Compute lim_{n → +∞} S_n.
Input :

Output :

Exercise 2
Suppose S_n=∑_k=1ⁿ k³/n⁴.
Compute lim_{n → +∞} S_n.
Input :

Output :

Exercise 3
Compute lim_{n → +∞}(1/n+1+1/n+2+...+1/n+n).
Input :

Output :

Exercise 4
Suppose S_n=∑_k=1ⁿ 32n³/16n⁴−k⁴.
Compute lim_{n → +∞} S_n.
Input :

Output :