Intereting Posts

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Irreducible $f(x) \in F$ of prime degree, $E/F$ finite extension, $p \mid $.
Why do statisticians like “$n-1$” instead of “$n$”?
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If $a^3+b^3+c^3=3$ so $\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a}\geq\frac{3}{2}$
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Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$
$\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$.
proving lines are perpendicular

Take a function that is analytic at 0 and consider its Maclaurin Series. Here are some examples I’ll refer to:

$$\frac{1}{1-x} =\sum_{n=0}^\infty x^n$$

$$\frac{1}{1+x^2} =\sum_{n=0}^\infty(-1)^nx^{2n}$$

$$\ln(1-x) =-\sum_{n=1}^\infty\frac{x^n}{n}$$

$$\sqrt{1-x} =1-\sum_{n=1}^\infty\frac{(2n-2)!}{2^{2n-1}n!(n-1)!}x^n$$

Each of these series has a radius of convergence of 1. And each function either

- Proving $\,f$ is constant.
- Proving $f'(1)$ exist for $f$ satisfying $f(xy)=xf(y)+yf(x)$
- Global invertibility of a map $\mathbb{R}^n\to \mathbb{R}^n$ from everywhere local invertibility
- Cannot calculate antiderivative using undefined coefficients
- Why can you mix Partial Derivatives with Ordinary Derivatives in the Chain Rule?
- Calculus of variations with two functions and inequality

$\bullet$ has a singularity along the edge of the disk of convergence (at 1, $\pm i$, and 1 in the first three examples respectively) or

$\bullet$ has a derivative with a singularity along the edge of the disk of convergence (the last example is this way at 1).

My question is: Suppose a function $f$ is analytic at 0 and its Maclaurin Series has a radius of convergence $r<\infty$. Does it have to be the case that some derivative (0th, 1st, 2nd, …) of $f$ blows up somewhere along the edge of the disk of convergence?

- integral to infinity + imaginary constant
- Convergence of $\sum_{n=1}^{\infty }\frac{a_{n}}{1+na_{n}}$?
- Basic Taylor expansion question
- Does $\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}}$ converge?
- Procedure for 3 by 3 Non homogenous Linear systems (Differential Equations)
- If $|f|$ is constant, so is $f$ for $f$ analytic on a domain $D$.
- Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?
- Evaluating $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_{2n}}{n}$
- Is arithmetic with infinite numbers fictitious?
- Uniform convergence of real part of holomorphic functions on compact sets

$f(x)=\sum x^n/e^{\sqrt n}$ has radius of convergence 1, and it and all its term-by-term derivatives converge everywhere on the unit circle. Basically, $e^{\sqrt n}$ goes to infinity faster than any polynomial but more slowly than any exponential.

- Finding Galois group of $x^6 – 3x^3 + 2$
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- Are values in a probability density function related to standard deviation?
- Show that $f(x) = x$ if $f(f(f(x))) = x$.
- Irreducible polynomial over Dedekind domain remains irreducible in field of fractions
- Is the order of universal/existential quantifiers important?
- Determining the Orbits/Orbit Space of $O(3)$ on Real 3 by 3 Traceless Symmetric Matrices
- Expectation of Ito integral, part 2, and Fubini theorem