Limit of Summation: Find Solution

[Kurani]

Homework Statement

Find the limit n to Infinity of summation (i=1 to n) 1/n * ((i/n)^2)

The Attempt at a Solution

I thought it was zero at first because 1/n goes to zero but apparently that's not right. I also tried to convert to an integral and got integral of i^2/n^2 which equals i^3/3*n^2 but that's not right either.

 

[LCKurtz]

Homework Statement

Find the limit n to Infinity of summation (i=1 to n) 1/n * ((i/n)^2)

The Attempt at a Solution

I thought it was zero at first because 1/n goes to zero but apparently that's not right. I also tried to convert to an integral and got integral of i^2/n^2 which equals i^3/3*n^2 but that's not right either.

Assuming you have typed what you meant to type, you can factor out the n's:

$$\sum_{i=1}^n \frac 1 n \frac {i^2}{n^2} = \frac 1 {n^3}\sum_{i=1}^n i^2$$

Do you know the formula for the sum of the first n squares? Put that in and see what happens as n → ∞.

 

[Kurani]

Yeah, I realized I had to do that right after I posted, it comes out to 1/3. Thanks