Uniform and pointwise convergence of sequence of function of $f_n(x) = [\log(1+x)]^n$

Question

I need to study the pointwise convergence of

$f_n(x) = [\log(1+x)]^n$

for every $x$ of the domain of the functions.

After i have to prove that the sequence of functions $f_n(x)$ is uniformly convergent to $f$ on the interval $[\frac{1}{2},1]$.

I've shown that in $(-1,\infty)$

$$dom(f_n(x))=\{x:x>-1\}$$

so for $x=0$ $f_n(x)=0 \xrightarrow{} 0$ for n $\rightarrow$ + $\infty$

It remains to show what happen for $x\neq 0$ but i don't find the pointwise convergence. How can i proceed? Thanks in advance for any help.

Answer 1

@andrew The pointwise limit is clearly $0$ for $x\in(-1+e^{-1},e-1)$. On $[1/2,1]$ you have $$ |\log(x+1)^n-0|=|\log(1+x)|^n<(\log 2)^n\to 0 $$ and this establishes the uniform convergence to zero.