Intereting Posts

Do we gain anything interesting if the stabilizer subgroup of a point is normal?
The functional equation $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$
Finding the area of a implicit relation
Cylinder in 3D from five points?
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$
How to prove that if a normed space has Schauder basis, then it is separable? What about the converse?
How to understand the regular cardinal?
Existence of acyclic coverings for a given sheaf
Proving that unconditional convergence is equivalent to absolute convergence
Cohomology easier to compute (algebraic examples)
Prove that a connected space cannot have more than one dispersion points.
Why is “working in $\mathbb {Z}_m$” essentially the same as “working with congruences modulo m”?
Math Major: How to read textbooks in better style or method ? And how to select best books?
Different definitions of trigonometric functions
Restriction of a Lebesgue integral to a subset of a measurable set.

How to find the radius of convergence of $\sum z^{n!}$?

I’m used to applying the ratio test to power series of the form $\sum a_{n}z^{n}$, but for a different power of $z$, I am a bit stumped. What about $\sum z^{2n+a}$ for another example? Where $a\in \mathbb{R}$.

- How to generalize Reshetnikov's $\arg\,B\left(\frac{-5+8\,\sqrt{-11}}{27};\,\frac12,\frac13\right)=\frac\pi3$?
- If $f,g$ are both analytic and $f(z) = g(z)$ for uncountably many $z$, is it true that $f = g$?
- Where does one use holomorphicity in the proof of Goursat's theorem?
- For what values $\alpha$ for complex z $\ln(z^{\alpha}) = \alpha \ln(z)$?
- A ‘strong’ form of the Fundamental Theorem of Algebra
- How to construct this Laurent series?

- Complex Analysis, Entire functions
- Which conformal maps UHP$\to$UHP extend continuously to the closure?
- Convergence or divergence of $\sum_{k=1}^{\infty} \left(1-\cos\frac{1}{k}\right)$
- Is the product of two non-holomorphic function always non-holomorphic?
- Unit ball in $C$ not sequentially compact
- What is a good complex analysis textbook?
- How to find partial sum of series and prove convergence?
- Evaluate integral: $ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx$
- If $f$ is a meromorphic modular form of weight $k$, then $\frac{1}{f}$ is a modular form of weight $-k$
- Asymptotic analysis of the integral $\int_0^1 \exp\{n (t+\log t) + \sqrt{n} wt\}\,dt$

HINT:

If $|z| = 1$, we know its behavior. If $|z| > 1$, then it explodes. And if $|z| < 1$, then $z^{n!} < z^n$, where I assumed $z$ was positive.

For the other one, note that you can factor out the $z^a$ to the outside of the sum, and are left just with $(z^2)^n$ on the inside.

Recall that the radius of convergence $R$ of the series $\sum\limits_na_nz^n$ is such that for every positive real number $r>R$, the real valued sequence $(x^{(r)}_n)$ defined by $x^{(r)}_n=|a_n|r^n$ is unbounded and for every positive real number $r<R$, this same sequence $(x^{(r)}_n)$ is bounded.

This dichotomy determines $R$ uniquely but of course much more is true since, for every $r<R$, $x^{(r)}_n\to0$ exponentially fast. On the other hand, the possible behaviours of $(x^{(R)}_n)$ (that is, at the critical value $r=R$) are more diverse since one can observe anything between (non exponential) convergence to zero and (non exponential) unboundedness.

*Application:* Consider any complex valued sequence $(a_n)$ such that $|a_n|\in\{0,1\}$ for every $n$ and introduce the set of indices $N=\{n\mid|a_n|=1\}$. Then the radius of convergence of the series $\sum\limits_na_nz^n$ is $R=+\infty$ if $N$ is finite, and $R=1$ if $N$ is infinite.

- What are the Eigenvectors of the curl operator?
- What is the mistake in doing integration by this method?
- Prove that $\lambda(A + B) \ge \lambda(A) + \lambda(B) $ for A and B being half-open intervals
- Separated schemes and unicity of extension
- Motivation behind the definition of localization
- Stolz-Cesaro Theorem, 0/0 Case
- How many ways can $r$ nonconsecutive integers be chosen from the first $n$ integers?
- Commutator Identities in Groups
- Learning Roadmap for Algebraic Topology
- What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?
- Eigenvalues of the circle over the Laplacian operator
- Showing when a permutation matrix is diagonizable over $\mathbb R$ and over $\mathbb C$
- Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.
- Point of maximal error in the normal approximation of the binomial distribution
- Finding all homomorphisms between two groups – couple of questions