List of top Mathematics Questions on Uniform Convergence asked in CPET

Consider the following statements:
(I) The sequence \(\{f_n\}_{n\ge1}\) of functions defined by \(f_n(x)=x^n\) is both point-wise and uniformly convergent on \([0,1]\).
(II) The sequence \(\{f_n\}_{n\ge1}\) of functions defined by \(f_n(x)=\dfrac{\ln(1+nx)}{n}\) is both point-wise and uniformly convergent on \([0,1]\).
Choose the correct answer.

Which of the following series does not converge uniformly on \((0,1)\)?
(I) \(\displaystyle\sum_{n=1}^\infty \frac{x^n}{n+x}\)
(II) \(\displaystyle\sum_{n=1}^\infty \frac{x}{n^2+x}\)
(III) \(\displaystyle\sum_{n=1}^\infty \frac{x^n}{1+x^n}\)

Consider the following statements:
(I) If \(a_n \ge 0\) for each \(n \in \mathbb{N}\) and the series \(\sum_{n=1}^\infty a_n\) converges, then the series \(\sum_{n=1}^\infty \dfrac{\sqrt{a_n}}{n^p}\) converges for \(p > 1/2\).
(II) The series \(\sum_{n=1}^\infty (-1)^n \sin(1/n)\) is conditionally convergent.
Which one of the options given below is correct?