I'd appreciate a double check of my work and logic, and some help answering some questions.
1.) Finding where {$f_n(x)$} converges, $f_n(x)=x(1-x^n)$
Using the ratio test:
$$L=lim_{n\to \infty} \sqrt {\frac {x-x^{n+1}}{{x-x^n}}}=\left|x\right|$$
Now with this test, $|x|\lt 1$ converges, and $|x|\gt 1$ diverges and if |x|=1 we need to do some further testing.
If I plug $x=\pm 1$ we get that $f_n$ converges on 1, but not -1.
Therefore the final answer would be on (-1,1].
EDIT: the above answer using the ratio test seems to be incorrect. I've left it up so I could get some advice as to where I went wrong (besides mucking up the algebra on the ratio test!)
2.) The next question is if the function converges uniformly on [0,1]. Checking point-wise convergence on [0,1] we see that $f_n(x)$ converges to 0. I'm not really sure how to test for uniform convergence, some help would be appreciated.
3.) Final question is if the convergence is "on average" on [0,1]. Finding a true translation rather than a direct translation proved difficult for this type of translation, but I believe we use the following test:
$$d_1(f_n,f)=\int_a^b|f_n-f|$$
What do we plug in for f to test it?
I appreciate any and all help.