For every \(n \in \mathbb N\), let \(f_n(x)=\frac{1}{x^n}\) for \(x\in[1,\infty)\). Show that \(\{f_n\}_{n\in \mathbb N}\) converges pointwise on \([1,\infty)\), and determine whether convergence is uniform on \([1,\infty)\). For a fixed \(R>1\), determine whether convergence is uniform on \([R,\infty)\).
For every \(n \in \mathbb N\), let \(f_n(x)=\arctan(nx)\) for \(x\in \mathbb R\). Show that \(\{f_n\}_{n\in\mathbb N}\) converges pointwise on \(\mathbb R\) to \[f(x) = \begin{cases} \pi/2, & \quad \text{if } x > 0.\\ 0, & \quad \text{if } x =0.\\ -\pi/2, &\quad \text{if } x < 0.\ \end{cases}\] Is the convergence uniform?
Show that for any \(\rho>0\) the sequence \(\bigl\{\frac{1}{nz}\bigr\}_{n\in\mathbb N}\) converges uniformly on \(\{\,z \in \mathbb C: \ {\left|z\right|}\geq \rho\,\}\).
Does \(\bigl\{\frac{1}{nz}\bigr\}_{n\in\mathbb N}\) converge uniformly on \(\mathbb C^\ast:=\mathbb C\setminus \{ 0\}\)?
For any \(\rho > 0\), show that \(\bigl\{\frac{n}{1+nz}\bigr\}_{n\in\mathbb N}\) converges uniformly on \(\{\,z \in \mathbb C: \ {\left|z\right|} > \rho \,\}\). Does it converge uniformly on \(\mathbb C^{\ast}\,\)?
Show that if \(0<\rho<1\), then the sequence \(\bigl\{\frac{1}{1+z^n}\bigr\}_{n\in\mathbb N}\) converges uniformly on \(\{\,z \in \mathbb C: \ {\left|z\right|}\leq \rho\,\}\) to the constant function \(f(z)=1\). On the other hand, show that the sequence converges uniformly on \(\{\,z \in \mathbb C: \ {\left|z\right|}\geq \rho^{-1}\,\}\) to the constant function \(f(z)=0\).
Show that the sequence \(\bigl\{\frac{1}{1+z^n}\bigr\}_{n\in\mathbb N}\) does not converge uniformly on \(\mathbb{D} = \{\,z \in \mathbb C: \ {\left|z\right|} < 1\,\}\).
For every \(n \in \mathbb N\), let \(f_n(z)=\sin(z/n)\) for \(z\in \mathbb C\). Show that \(\{f_n\}_{n\in\mathbb N}\) converges pointwise on \(\mathbb C\). Let \(\rho\) be a positive real number. Show that \(\{f_n\}_{n\in\mathbb N}\) converges uniformly on \(\{z:{\left|z\right|}\le \rho\}\). Show that \(\{f_n\}_{n\in\mathbb N}\) does not converge uniformly on \(\mathbb C\).
For every \(n \in \mathbb N\), let \(f_n(x)=\cos\left(1+\frac{x}{n}\right)\) for \(x\in\mathbb{R}\). Show that \(\{f_n\}_{n\in\mathbb N}\) converges pointwise and determine whether convergence is uniform on \(\mathbb{R}\). For fixed \(R>0\), is the convergence uniform on \([0,R]\)?
Show that the series
\(\displaystyle
\sum_{k=1}^\infty\frac{2^kz^{2k}}{k^2}\) converges uniformly on
\(%\bar B_{\frac{1}{\sqrt 2}}(0) =
\left\{ z \in \mathbb C: \: |z|\leq\frac{1}{\sqrt{2}}\right\}\),
and deduce that the limit function is continuous on this set.
Prove that \(\sum_{n=0}^{\infty} e^{nz}\) converges uniformly on \(\{z \in \mathbb C: \: \mathop{\text{Re}}(z)\le -1\}\), but not on \(\{z \in \mathbb C: \: \mathop{\text{Re}}(z)\le 0\}\).
Let \(R\) satisfy \(0<R<1\). Show that the series \(\displaystyle\sum_{n=1}^{\infty}\frac{z^{n}}{1+z^n}\) converges uniformly on \(%B_R(0)= \{z \in \mathbb C: \: |z|<R \}\). Conclude that the infinite series defines a continuous function on the unit disc \(\mathbb D\).
Prove that each of the following series converge uniformly on the corresponding subset of \(\mathbb C\): \[\begin{aligned} (a)&\ \sum_{n=1}^{\infty}\frac{1}{n^2z^{2n}}, &{\rm on}&\quad \{\,z \in \mathbb C: \: {\left|z\right|}\geq 1\,\}.\\ (b)&\ \sum_{n=1}^\infty\sqrt n \, e^{-nz}, &{\rm on}&\quad \{\,z\in \mathbb C: \: \ 0<r\leq \mathop{\text{Re}}(z)\,\}.\\ (c)&\ \sum_{n=1}^\infty\frac{2^n}{z^n+z^{-n}}, &{\rm on}&\quad \left\{\,z\in \mathbb C: \: \ {\left|z\right|}\leq r<\frac{1}{2}\right\}.\\ (d)&\ \sum_{n=1}^\infty2^{-n}\, \cos(nz), &{\rm on}&\quad \{\,z\in \mathbb C: \: \ {\left|\mathop{\text{Im}}(z)\right|}\leq r<\ln 2\, \}. \end{aligned}\]
Given \(0<r<R<\infty\), show that \(\displaystyle\sum_{n=1}^{\infty}\frac{\left(z+\frac{1}{z}\right)^n}{n!}\) converges uniformly on \(r < |z| < R\). Conclude that the infinite series defines a continuous function on \(\mathbb C^{\ast}\).
Prove that \(\displaystyle{\sum_{n=1}^\infty \frac{z^n}{1+z^{2n}}}\) converges uniformly on \(|z|< r\), for any \(r<1\). Prove it also converges uniformly on \(|z|\ge R\), for any \(R>1\). Conclude that the infinite series defines a continuous function inside and outside the unit circle. What is the situation on the unit circle?