The series $$\sum_{n=1}^\infty\arccos{{n^4x^4}\over{1+n^4x^4}}$$ supposedly converges uniformly on any interval $I$ for which $0\notin\overline{I}$ while $$\sum_{n=1}^\infty\arccos{{n^2x^2}\over{1+n^2x^2}}$$ does not converge at all for any $x$.
Firstly, how should I go about proving this? The derivatives of the summands don't seem to converge... If there were say $\arctan$ instead of $\arccos$, I could use $|\arctan x| \leq |x|$... If there were other terms, I could use the boundedness... But I don't know how to estimate the arccosines... Moreover, can $I$ really be unbounded? What if I chose $x_n = {{1}\over{n}}$...
Thank you.
Let $a>0$ and let's prove the uniform convergence of the first series on $[a,+\infty)$. Let
$$g_n(x)=\frac{n^4x^4}{1+n^4x^4}$$ then by the derivative we get easily that $g_n$ is increasing and then $f_n=\arccos(g_n)$ is decreasing then
$$||f_n||_\infty =f_n(a)$$ Moreover, we have
$$\cos(f_n(a))=g_n(a)\sim_\infty 1-\frac1{x^4 n^4}\implies f_n(a)\sim_\infty\frac{\sqrt2}{x^2n^2}$$ and since the series $\sum\frac1{n^2}$ is convergent so the given series is uniformly convergent on $[a,+\infty)$.