Let $f_n(x) =a x^n +b $cos$(\frac {x}{n})$. Find $a,b \in \bf R$ such that $f_n(x)$ converges uniformly.
I think that for existence of point wise limit of $f_n$ we've to assume that $a=0$ for $ \vert x \vert >1$ and then point wise limit is $b$ .For $ \vert x \vert <1$,again the limit is $b$ but we don't assume here that $a=0$.For $x=1$ ,limit is $a+b$ and at $x=-1$ limit does not exist.Is this correct ?How to proceed further?Do we need to consider separate cases ?