I have $\sum_{n=1}^{+\infty} (1-\frac{n}{x}sin(\frac{x}{n}))^\alpha$ with $\alpha \in R$.
I have found pointwise convergence set $E=R-\{-0\}$ if $\alpha>1/2$.
For uniformly convergence I want calculate Sup$_E$ |$f_{n,\alpha}(x)|$.
$f_{n,\alpha}(x)$ are pair so I consider $x>0$. With the substitution $t=x/n$ I have Sup$_{t>0} |1-\frac{sint}{t}|^{\alpha}$.
Is this $Sup >1$?