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Apr
25
comment Show that a power series is analytic inside its radius of convergence
I really like your last explanation - at least it is more formal than what I had in mind thus far. What do you think of the addition that we are looking for variables $z$ close to our new center $z_0$ and thus using the triangle inequality holds for those cases? Prosaic: your example would be false in general as your mentioned point $z=3$ wouldn't be in the circle as it is too small since the outer radius isn't allowing it to be bigger.
Apr
24
comment $(x-x_0)^0$ in power series
Whoever just downvoted me: would you be so kind as to explain why? Maybe I can improve my answer.
Apr
24
comment Holomorphic, entire functions and the Cauchy-Riemann equations
@Ant So ignoring $c_1$ and $c_2$ in 2. I am getting $f(z)=-\mathrm iz^2$ which seems to be correct - I assume you started with a polynomial of degree 2 since there are no higher degrees in my representation of $f(x+\mathrm iy)$, didn't you?
Apr
24
comment Holomorphic, entire functions and the Cauchy-Riemann equations
@Ant So it is only $\mathbb C$-differentiable at $z=0$ but there is no holomorphic restriction, then?
Apr
24
comment Show that a power series is analytic inside its radius of convergence
My newest result is that $f(z)$ converges for $|z|<R$ and via the triangle inequality we get $|(z-z_0)+z_0|<R\implies |z-z_0|<R-|z_0|$. Furthermore we know (based on $f(z)<\infty$ for those $z$) that $\sum_{n=0}^{\infty}\sum_{k=n}^\infty\binom{n}{k}a_nz_0^{n-k}(z-z_0)^k<\infty$. Thus again via the triangle inequality the absolute value of that sum is $<\infty$ as well.
Apr
24
comment Show that a power series is analytic inside its radius of convergence
I have a follow-up question on the first solution especially as I have an exercise which is based on a power series $f(z)=\sum_{n=0}^\infty a_nz^n$ and $|z_0|<R$ where $R$ is the radius of convergence of $f(z)$. I have to show that for $|z-z_0|<R-|z_0|$ we get $\sum_{n=0}^\infty\sum_{k=0}^n\binom{n}{k}|a_n||z_0|^{n-k}|z-z_0|^k<\infty$. I could get this expression with my application of the binomial theorem but I don't have an explanation why it converges for $|z-z_0|<R-|z_0|$ - could you give me some suggestions on that one?
Apr
21
comment Determining limit points and proving there are no more of them
@AlexanderCska Let $n_k=2k$ then you get multiple limit points in the first case as $\sin(2k\pi/4)\in\{-1,0,1\}$ based on $k$ and this is not a convergent sub-sequence.
Apr
16
comment When is $\sin\colon\mathbb{C}\to\mathbb{C}$ purely real/imaginary?
@stochasticboy321 So assuming $y=0$ purely real for all $x$ and for $y\neq 0$ purely imaginary for $x=n\pi,n\in\mathbb Z$ as the real component vanishes, isn't it? Hence purely imaginary on $\{(x,y)\mid x=n\pi,y\neq 0,n\in\mathbb Z\}$ while purely real on $\{(x,0)\mid x\in\mathbb R\}$.
Apr
15
comment Calculating exponential limit
First of all What have you tried thus far and why do you think you know the result? and furthermore did you mix up $n$ and $x$ variables? What is $x$?
Apr
1
comment Simplify $(n+1)^{1/\sqrt{n}}$ and $(n^2+1)^{1/n}$
Yeah I was considering $\sqrt[n]{n}\to 1$ as well but couldn't really compare the sequences to this root.
Mar
30
comment Issues computing $\int\cos^2x~\mathrm{d}x$ without using $\sin^2(x)=1-\cos^2(x)$
I know that I am going round and round but I was curious whether there is a more thorough explanation why besides $(\cos x)''=-\cos x$.
Mar
29
comment Issues computing $\int\cos^2x~\mathrm{d}x$ without using $\sin^2(x)=1-\cos^2(x)$
@YuriyS Chosing between $\sin x$ and $\sin x$ can yields different results - it the same :P
Mar
26
comment Prove the series $\sum_{n=1}^∞ (-1)^n(n)/(n+2)$ diverges
@Garrett $$\frac{n}{n+2}=\frac{1}{1+2/n}\overset{n\to\infty}{\longrightarrow}1\neq 0$$
Feb
12
comment Deducing the series expansion of $\arctan(x^2)$ via the series expansion of $\arctan(x)$ at $x=0$
So we must show that the replacement still yields a valid power series and that it has the same radius of convergence?
Jan
27
comment Multivariable taylor series expansion of $\exp(-(x^2+y^2))$
Am I right in assuming that $-\exp(-5)(4x+2y-11)$ would be the right polynomial in my case?
Jan
21
comment Eigenvectors of $\begin{pmatrix}6&2\\-10&-1\end{pmatrix}$ (linear equations with complex numbers)
@Prospect I do know the algorithm but I am struggling with the particular steps in this specific case.
Dec
9
comment Continuous function taking each of its values twice
It seems I am misunderstanding something in regards to the claim from the paper as all examples that come to mind have a finite amount of discontinuities.
Dec
9
comment Continuous function taking each of its values twice
Every function that satisfies this property has an infinite number of discontinuities. Refer to ams.org/journals/proc/1986-098-02/S0002-9939-1986-0854049-8/…
Dec
9
comment Using L Hopital's Rule Using $e$
$e=\exp(1) \rightarrow (\exp(1))'=(1)'\cdot\exp(1)=0$ as AndreasT mentioned.
Dec
8
comment Definite integral $\int _{0}^{1}\sqrt [3] {2x^{3}-3x^{2}-x+1}~\mathrm dx$
What have you tried so far?