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In the following cases examine whether the sequence $\{f_n\}_{n\in\mathbb{N}}$ of complex functions converges uniformly or not: $f_n(z)=z^n\,(1-i\,z)^n\,, \quad |z|<1$, $f_n(z)=\dfrac{{\rm{e}}^{-n\,\Re(z)\,i}}{n\,|z|}\,, \quad z\in\mathbb{C}\setminus\{0\}$, $f_n(z)={\rm{Arg}}\,\big(\frac{z}{n\,\over...

Examine whether the sequence of functions $g_n:[0, 1] \longrightarrow \mathbb{R}$ defined as \[g_n(x)=\begin{cases}\dfrac{x^{\frac{1}{n}}\log(1+x)}{x\,(1+x^{\frac{2}{n}})^{\frac{3}{2}}}\,,& x\in(0,1]\\ 0\,,& x=0\end{cases}\,,\quad n\in\mathbb{N}\,,\] converges uniformly on $[0, 1]$ or not. ...

Evaluate \[\displaystyle\int_{-1}^1 \log(1-x)\,\log(1+x)\, dx\,.\]

Evaluate the convergent series \[\mathop{\sum}\limits_{n=2}^{\infty} \frac{(-1)^n}{n\log{n}}\,.\]


Note: I don't have a solution for this.

i. For $x\in(0,1)$ we have that \begin{align*} \displaystyle\mathop{\lim}\limits_{n\to+\infty}f_n(x)&=\mathop{\lim}\limits_{n\to+\infty}\sqrt{n} \,x^2 \,( 1 - x^2)^n\\ &=x^2\mathop{\lim}\limits_{n\to+\infty}\frac{\sqrt{n}}{( 1 - x^2)^{-n}}\\ &\stackrel{\frac{\infty}{\infty}}{=}x^2\matho...

We'll prove something more general: If $f:U\subseteq{\mathbb{R}}^3\longrightarrow{\mathbb{R}}$ is a twice differentiable function in some open set $U\subseteq{\mathbb{R}}^{3}$, then $\nabla\times\big(\nabla {f}\big)\equiv \overline{0}$ in $U$. Proof: \begin{align*} \nabla{f}&= \bigg(\frac{\parti...

Let \(\color{red}{x\in\mathbb{R}}\). Prove that the sequence \(\displaystyle{s_n=\sin x+\dots+\sin(n\,x)\,,\; n\in\mathbb{N}}\) is bounded. I think there is a misunderstanding here about boundedness. Let me try to elucidate. If we consider $x$ as a fixed real number, then the sequence $s_n(x)$ is a...

For $x\neq2m\pi\,,\; m\in\mathbb{Z}$, maybe the formula \[s_n=\mathop{\sum}\limits_{k=1}^n\sin(kx)=\frac{\sin\big(\frac{(n+1)\,x}{2}\big)\,\sin\big(\frac{nx}{2}\big)}{\sin\big(\frac{x}{2}\big)}\] and that \[|s_n|=\Bigg|\frac{\sin\big(\frac{(n+1)\,x}{2}\big)\,\sin\big(\frac{nx}{2}\big)}{\sin\big(\fra...

PJPu17 wrote:I can´t do this, because i have the metric induce by the euclidean ( the first fundamental form on the sphere), and these fields are in cartesians. :S

Which are the charts for the sphere that you are considering?

...Is there any geometric argument to prove the statement without doing any operation? Using only the compatibility of the connection with the metric, and the orthogonality of both fields in cartesians? Maybe you're right! But I have no answer about this. My suggestion is: write $X(t)=\mathop{\sum}...