Let s(n) = [sigma i = 1 to n ] ( 1 + i/n)^2 (1/n)?

Question

The expression is read as
(1 + i divided by n) raised to the second power
times (1/n)
What is the limit as n goes to infinity?

Answer 1

This is a Riemann sum for the integral of x^2 from x=1 to x=2. An anti-derivaqtive of x^2 is x^3/3. Evaluate this at x=1 and x=2 and subtract to get 7/3.

Answer 2

Mathematica agrees with MatCad, 7/3. It evaluates the sum to ((1+2n)*(1+7n))/6n^2 Of which you can calculate the limit as n goes to infinity.

Answer 3

Mathcad gives answer of 7/3

Answer 4

O MY BRAIN............i give up.