Sigma notation question HELP!!!

[scorks]

Hey, I'm doing sigma notation, and the question is "if a limit diverges to infinity, state whether it diverges to positive or negative infinity" . Now, I recall doing some of this a few years ago in a pre-calc class that I did, but I'm just not confident I'm doing it right. The question and my workings are [below] So, my final answer was 0... The problem is, ALL the questions have a (1/n) outside of the sigma, so doesn't that mean they'll ALL come out to be 0?

\(\displaystyle \mbox{d. }\, \displaystyle{\lim_{n \to \infty}}\, \dfrac{1}{n^3}\, \sum\limits_{i=1}^n\, \left(1\, -\, i^3\right)\)

\(\displaystyle =\, \dfrac{1}{n^3}\, \sum\limits_{i=1}^n\, (1)\, -\, \dfrac{1}{n^3}\, \sum\limits_{i=1}^n\, (i^3)\)

\(\displaystyle =\, \left(\dfrac{n}{n^3}\right)\, -\, \left(\dfrac{n^2(n\, +\, 1)^2}{4n^3}\right)\)

\(\displaystyle =\, \dfrac{\left(\dfrac{n}{n}\right)}{\left(\dfrac{n^3}{n}\right)}\, -\, \dfrac{\left(\dfrac{n^2}{n^2}\right)\left(\dfrac{(n\, +\, 1)^2}{n^2}\right)}{\left(\dfrac{4n^3}{n^2}\right)}\)

\(\displaystyle \rightarrow \, \dfrac{1}{\infty}\, -\, \dfrac{(1)\left(\dfrac{(n\, +\, 1)^2}{n^2}\right)}{\infty}\)

 

[DrPhil]

_{Hey, I'm doing sigma notation, and the question is "if a limit diverges to infinity, state whether it diverges to positive or negative infinity" . Now, I recall doing some of this a few years ago in a pre-calc class that I did, but I'm just not confident I'm doing it right. The question and my workings are in the picture. So, my final answer was 0... The problem is, ALL the questions have a (1/n) outside of the sigma, so doesn't that mean they'll ALL come out to be 0?}View attachment 3237

Good start. The first sum goes to 1/n^2 ---> 0, so you have

\(\displaystyle \displaystyle \lim_{n \to \infty}\dfrac{-n^2(n+1)^2}{4n^3} = - \frac{1}{4}\lim_{n \to \infty}\dfrac{(n+1)^2}{n}\)

When n gets large, you can ignore +1 as being very small compared to n. Then you have n^2/n = h ---> infinity. The sign is the minus sign out front.

Your error was when you divided numerator and denominator by n^2, that you divided the numerator twice [as if there were two terms instead of two factors].