Intereting Posts

Definition of a Functor of Abelian Categories
Proof of $M$ Noetherian if and only if all submodules are finitely generated
In which cases are $(f\circ g)(x) = (g\circ f)(x)$?
What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$
Showing $x^3$ is not uniformly continuous on $\mathbb{R}$
Efficiently calculating the logarithmic integral with complex argument
Is the function $f(x)=|x|^{1/2}$ Lipschitz continuous?
second derivative of the inverse function
A probability measure that takes only the values $0$ and $1$ but is not a point mass
Semisimple rings
Find a formula for all the points on the hyperbola $x^2 – y^2 = 1$? whose coordinates are rational numbers.
Klein-bottle and Möbius-strip together with a homeomorphism
Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$
Product rule intuition
morphism from a local ring of a scheme to the scheme

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

- How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$
- Are these two functions equal?
- How does the existence of a limit imply that a function is uniformly continuous
- Integration by substitution gone wrong
- Improve my proof about this $C^\infty$ function even more!
- Prove that $x^2<\sin x \tan x$ as $x \to 0$

$\bullet$ 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?

- What is the derivative of a vector with respect to its transpose?
- Evaluating $\int_a^b \frac12 r^2\ \mathrm d\theta$ to find the area of an ellipse
- Why does $\lim_{n\to\infty} z_n=A$ imply $\lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A$?
- Infinite product: $\prod_{k=2}^{n}\frac{k^3-1}{k^3+1}$
- Prove that $\arctan\left(\frac{2x}{1-x^2}\right)=2\arctan{x}$ for all $|x|<1$, directly from the integral definition of $\arctan$
- Is $ \frac{\mathrm{d}{x}}{\mathrm{d}{y}} = \frac{1}{\left( \frac{\mathrm{d}{y}}{\mathrm{d}{x}} \right)} $?
- What is the equation for a line tangent to a circle from a point outside the circle?
- Question about quadratic equation of complex coefficients.
- How to prove that $\lim_{x \to \infty} \frac{x^2}{2^x}=0$
- Prove using contour integration that $\int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$

$f(x)=\sum x^n/e^{\sqrt n}$ 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.

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- How to find $\lim_{(x,y)\rightarrow (0,0)} \frac{\sin(x\cdot y)}{x}$?
- Infinite Series using Falling Factorials
- A finite field cannot be an ordered field.
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- Variable in Feynman Integration
- The units of $\mathbb Z$
- Equivalent way of writing the norm of Lp
- A property of exponential of operators
- Is $\{\varnothing \}$ an empty set?
- In how many ways can we put $31$ people in $3$ rooms?
- Prove that $Q_8 \not < \text{GL}_2(\mathbb{R})$
- Estimation of sums with number theory functions