13.3.4  Riemann sums

Given a function f on [0,1], the Riemann sum corresponding to dividing the interval into n equal parts and using the right endpoints is

The sum_riemann command determines if a sum is such a Riemann sum, and if it is, evaluates the integral.

• sum_riemann takes two arguments:
  • expr, an expression depending on two variables.
  • [n,k], the list of those two variables.
• sum_riemann(expr,[n,k]) returns

  lim n→+∞

  n ∑ k=1

  expr

  (which, viewing the sum as a Riemann sum of a continuous function on [0,1], is the definite integral) or returns "it is probably not a Riemann sum" when the no result is found.

Examples

Let S_n=∑_k=1ⁿ k²/n³. Compute the limit of (S_n)_n∈ℕ.

Let S_n=∑_k=1ⁿ k³/n⁴. Compute the limit of (S_n)_n∈ℕ.

Compute lim_{n → +∞}(1/n+1+ 1/n+2+⋯+1/2n).

Let S_n=∑_k=1ⁿ 32n³/16n⁴−k⁴. Compute the limit of (S_n)_n∈ℕ.