1. ## Integral Homework Question

Good Evening Everyone, recently in my first year single variable Calculus class we started learning integrals. I have been working on a homework problem that I believe to have solved as my prof said the answer was 0. Proof is posted and I would just like some clarification with the format of my answer (ie. is right). Thank you for your help.
The question is:

2. ## Re: Integral Homework Question

Your very first step is wrong. You have, to start,
$\displaystyle \frac{3}{n}\left[\sum_{n=1}^\infty \left(\frac{3i^2}\right){n^2}- 2\left(\frac{3i}{n}\right)\right]$
and the next line is
$\displaystyle \frac{3}{n}\left[\frac{1}{n^2}\sum_{n=1}^\infty 3i^2- 6i\right]$
You have factored $\displaystyle 1/n^2$ out of both first and second terms in the sum, but there was no "$\displaystyle 1/n^2$" in the second term.

Instead, divide it into two sums: $\displaystyle \frac{3}{n}\left[\frac{1}{n^2}\sum_{n=1}^\infty 3i^2- \frac{1}{n}\sum_{n=1}^\infty 6i\right]$

That is the same as $\displaystyle \frac{3}{n}\left[\frac{3}{n^2}\sum_{n=1}^\infty i^2- \frac{6}{n}\sum_{n=1}^\infty i\right]$
You should know formulas for the sums $\displaystyle i$ and $\displaystyle i^2$