6.20.4 Riemann sum: sum_riemann

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.

Exercises.

1. Suppose S_n=∑_k=1ⁿ k²/n³.
   Compute lim_{n → +∞} S_n.
   Input:
   sum_riemann(k^2/n^3,[n,k])
   Output:

   1 3

2. Suppose S_n=∑_k=1ⁿ k³/n⁴.
   Compute lim_{n → +∞} S_n.
   Input:
   sum_riemann(k^3/n^4,[n,k])
   Output:

   1 4

3. Compute lim_{n → +∞}(1/n+1+ 1/n+2+…+1/n+n).
   Input:
   sum_riemann(1/(n+k),[n,k])
   Output:

   ln

   ⎛ ⎝

   2

   ⎞ ⎠

4. Suppose S_n=∑_k=1ⁿ 32n³/16n⁴−k⁴.
   Compute lim_{n → +∞} S_n.
   Input:
   sum_riemann(32*n^3/(16*n^4-k^4),[n,k])
   Output:

   2 arctan

   ⎛ ⎜ ⎜ ⎝

   1 2

   ⎞ ⎟ ⎟ ⎠

   +ln

   ⎛ ⎝

   3

   ⎞ ⎠