Riemann sums

### 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

n
∑
k=1
| f | ⎛ ⎜ ⎜ ⎝ |
x
n
| ⎞ ⎟ ⎟ ⎠ |
1
n
| .

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 Sn=∑k=1n k2/n3. Compute the limit of (Sn)n∈ℕ.

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

1
3

Let Sn=∑k=1n k3/n4. Compute the limit of (Sn)n∈ℕ.

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

1
4

Compute limn → +∞(1/n+1+ 1/n+2+⋯+1/2n).

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

ln | ⎛ ⎝ | 2 | ⎞ ⎠

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

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

2 arctan | ⎛ ⎜ ⎜ ⎝ |
1
2
| ⎞ ⎟ ⎟ ⎠ | +ln | ⎛ ⎝ | 3 | ⎞ ⎠