Find the value of the expression.

Question

$$ \sum_{i=1}^{1000} i^\frac{-2}{3} =M$$

$$\text{Then find the value of [M]-20} $$ $$\text{where[ . ] Denotes the greatest integer function.}$$ My attempt, I was able to figure out the lower end of M as $$\sum_{i=1}^{1000} i^\frac{-2}{3} \gt \int_{1}^{1000} x^\frac{-2}{3} dx$$ which yielded the result that $$ M \gt 27$$ But now I struggle to setup an upper bound on this thing. Any help is appreciated.

score 3 · Accepted Answer · answered Oct 11 '19 at 15:41

3

$$\small\int_1^{1000}x^{-2/3}\,dx>\sum_{j=2}^{1000}(j-(j-1))\cdot j^{-2/3}\implies27>M-1\implies27<M<28\implies \lfloor M\rfloor-20=7.$$

answered Oct 11 '19 at 15:41

ə̷̶̸͇̘̜́̍͗̂̄︣͟

27,005

Can you please explain the logic behind (j-(j-1)) – DareDevil Oct 11 '19 at 15:44
Oh I get it! Thank you for your efforts. – DareDevil Oct 11 '19 at 15:47
Consider the rectangle with vertices at $(j-1,0)$, $(j,0)$, $(j-1,j^{-2/3})$ and $(j,j^{-2/3})$. This is below the curve $f(x)=x^{-2/3}$ and has area $(j-(j-1))(j^{-2/3})=j^{-2/3}$. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Oct 11 '19 at 15:47

Find the value of the expression.

1 Answers1