# Tagged Questions

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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### $x+2$ is irreducible in the power series ring $\mathbb{Z}[[x]]$

For the last few days I am trying to prove that $x+2$ is irreducible in $\mathbb{Z}[[x]]$. I think that it is false... I would be very much thankful for any kind of suggestions and help.
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### What function does this series represent?

A midterm I proctored recently showed that $$\cos(\sqrt{x}) = \sum_{k=0}^{\infty} \dfrac{(-1)^k x^k}{(2k)!}$$ The question asked what function this series represents. It may represent cosine, but ...
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### Power series divergence (real analysis)

Show that if a power series diverges at $x_0$ then it must also diverge when $\lvert x\rvert > \lvert x_0\rvert$ or provide a counterexample. I feel like there is a counterexample for some kind of ...
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### problem convergent power series expansion such that $f^{(n)}(x)$ and $|f^{(n)}(x)|\leq 1$ for every $n\geq 1$ and for every $x\in(-1,1)$

Let $f:(-1,1)$ $\to \mathbb{R}$ such that $f^{(n)}(x)$ exists and $|f^{(n)}(x)|\leq 1$ for every $n\geq 1$ and for every $x\in(-1,1)$. Then f has a convergent power series expansion in a neighbourhood ...
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### Non-trivial examples of power series which are uniformly convergent on $[0,1)$ and left-continuous at $x = 1$

The question is motivated by a more extensive problem that needs a formal proof, but I am not interested in help on the proof itself, but I'd like to see some examples of such power series. I put non-...
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### Prove $\lim\limits_{n \to \infty} \sup \left ( \frac{(2n - 1)^{2n - 1}}{2^{2n} (2n)!)} \right ) ^ {\frac 1 n} = \frac {e^2} 4$

This is a problem in Heuer (2009) "Lerbuch der Analysis Teil 1" on page 366. I assume that the proof should use $e = \sum\limits_{k = 0}^{\infty} \frac 1 {k!}$, but I cannot come further.
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### finding the radius of convergence of a complex power series

I am trying to find radius of convergence of $$\sum_{n=0}^{\infty} z^{a^n}$$ where $a>1$ integer. I obviously want to use $1/R = \limsup ( |c_n| )^{1/n}$. Is there a way to write $z^{a^n}$ ...
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### Divergent Sum Renormalisation

I noticed an interesting property of holomorphic functions and I'm wondering if it forms the basis of divergent sum renormalisation. Let $f,g:\mathbb C \rightarrow \mathbb C$ be holomorphic functions....
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