which of following are uniformly convergent?

Question

which of the following are uniformly convergent?

1. $\frac{1}{n(1+x^2)}$ in $\mathbb{R}$
2. $\frac{sin~nx}{n}$ in $\mathbb{R}$
3. $\frac{x^2+nx}{n}$ in $\mathbb{R}$
4. $\frac{x^n}{n}$ in [0,1]

All except 3 is pointwise convergent and converges to 0. But to check uniform convergence Wiestrauss M-test is not working as 1/n is not absolutely convergent. I think these three are uniformly convergent. How to prove this

Answer 1

For (4). It suffices to prove that the 'largest' separation from the terms in the sequence to the limit shrinks to zero to prove uniform convergence.

Let $f_n(x)=\frac{x^n}{n}$ and we consider the following supremum. $$sup\{|f_n(x)|:x\in[0,1], n\in\mathbf{N}\}\leq sup\{|f_n(1)|: n\in\mathbf{N}\}= 0 $$ $ so the sequence much converge uniformly. I recommend applying this method to the others.