Question

Sum of rectangles inequality: In the definition of the integral of x^2 between 0 and b, a geometric example was used. The largest square pyramid contained by the step pyramid has the formula 1/3*n^3, and the smallest square pyramid that would contain the step pyramid has the formula 1/3*(n+1)^3. This inequality ends up looking like 1/3 < (1^2 + 2^2 + 3^2 + n^2)/n^3 < 1/3*(n+1)^3/n^3. This all ends up being equal to 1/3*(1+1/n)^3. I am missing the intermediate step between the inequality and what it is equal to. Although I can visualize why, I do not know of any intermediate algebraic step.

Answer 1

\[1/3 \frac{(n+1)^3}{n^3}\]
\[=1/3 (\frac{n+1}{n})^3\]
\[=1/3(\frac{n}{n} + \frac{1}{n})^3\]
\[=1/3(1+\frac{1}{n})^3\]
Now if n is approaching infinity then 1/n will be 0 so the above will become equal to
just
\[\frac{1}{3}\]

Answer 2

Thank you so much.