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Feb
10
comment Writing a function $f : [-\pi,\pi) \to \mathbb{R}$ as $\sum c_k e^{ikx}$ where $c_k$ is to be found
$c_k=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{f(x)e^{ikx}\,dx} = \frac{1}{2\pi}\int\limits_{-\pi}^{0}{(-1)\cdot e^{ikx}\,dx}+\frac{1}{2\pi}\int\limits_{0}^{\pi}{1\cdot e^{ikx}\,dx}.$
Feb
10
comment Sum and radius of $\sum_{n=0}^{\infty}(\cos(5n)+i \sin(5n))z^{n}$
$|e^{i \varphi}|=1$ for all $\varphi \in \mathbb{R}.$
Feb
10
answered To evaluate $\lim_{x \to 0^+} \frac{\log(x)}{\sqrt x}$ using inequality
Feb
6
revised Evaluate the definite integral $\int_{0}^{a}\frac{dx}{(a^2+x^2)^{3/2}}$
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Feb
6
comment Evaluate the definite integral $\int_{0}^{a}\frac{dx}{(a^2+x^2)^{3/2}}$
Hint: Try using substitution $x=a\sinh{t}.$
Feb
5
reviewed Approve Find the following limits without using l'Hospital's rule
Feb
5
comment Find the following limits without using l'Hospital's rule
Changing of variables as $t=\frac{\pi}{4}-x$ in the first case and $t=1-x$ in the second is a helpful way. Combine it with appropriate trigonometric identities and the fact that $\lim_{x \to 0} \frac{\sin x}{x} = 1.$
Feb
3
comment $z^n=(i+z)^n$, solve for $z$
If $z=r(\cos{\varphi}+i\sin{\varphi})$ then $1+z=(1+r\cos{\varphi})+ir\sin{\varphi}.$
Jan
22
comment Showing convergence of the integral $\int_{1}^{\infty}\frac{\ln(x)\cos(x)}{x^2+1}\,{\rm d}x$
Hint: For an arbitrary fixed $\varepsilon>0 \ \lim\limits_{x\to\infty}{\frac{\ln{x}}{x^\varepsilon}}=0,$ which imply $\ln{x} < x^\varepsilon$ for all $x > x_\varepsilon.$
Jan
21
answered I would like to calculate limit: $ \lim_{n \to \infty}{n\cdot \ln({1-\arctan{\frac{3}{n}}})} $
Jan
20
reviewed Approve Proof of $a^{\frac{p-1}{2}} \equiv \left(\frac{a}{p}\right)$ mod $p$
Jan
20
comment Reducing a particular fraction
$\frac{t-1}{t} =1-\frac{1}{t} $
Jan
20
comment Determine if this power series is convergent for Z on the boundary of it's disc of convergence
Since the radius of convergence $R=2,$ then for each point on the boundary of disc of convergence we have $z+2i=2e^{i\varphi}.$ Thus for these points $$(1+i\sqrt{3})\sum\limits_{n=0}^{\infty}{\dfrac{1}{n^2 2^n}(z+2i)^n}=(1+i\sqrt{3})\sum\limits_{n=0}^{\infty}{\dfrac{2^n e^{in\varphi}}{n^2 2^n}}=(1+i\sqrt{3})\sum\limits_{n=0}^{\infty}{\dfrac{ e^{in\varphi}}{n^2}},$$ and the last series is absolutely convergent.
Jan
20
comment Determine if this power series is convergent for Z on the boundary of it's disc of convergence
You are welcome!
Jan
20
comment Determine if this power series is convergent for Z on the boundary of it's disc of convergence
Rewrite series as $(1+i\sqrt{3})\sum\limits_{n=0}^{\infty}{\dfrac{1}{n^2 2^n}(z+2i)^n}$ and denote $c_n=\dfrac{1}{n^2 2^n}.$ Then using the ratio test $R=\lim\limits_{n\rightarrow\infty} \left| \dfrac{c_n}{c_{n+1}} \right|=\lim\limits_{n\rightarrow\infty} \left| \dfrac{2^{n+1}(n+1)^2}{2^n n^2} \right|= 2 \lim\limits_{n\rightarrow\infty} {\left(1+\frac{1}{n}\right)^2}=2.$
Jan
20
comment Determine if this power series is convergent for Z on the boundary of it's disc of convergence
How you got the radius of convergence $R=1?$
Jan
13
comment Find $f(x)$ where $x\in[0,1], f(x) =\lim_{k\rightarrow \infty}\sum_{n=1}^{k}((1-x)^2x^n)$
Hint: Consider limit for cases $x \in [0,\,1)$ and $x=1.$
Jan
13
answered Unspecific boundaries for finding area by double integration
Jan
11
reviewed Approve Am I understanding this question correctly? Linear programing
Jan
11
reviewed Approve Probability of drawing a better card on the second go?