# 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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### One to one map $f$ equal to its power series

Across a difficult exercise sheet I encountered this exercise : Let $f$ be a continuous map from $\bar D$ the closed unit disk (in $\mathbb{C}$) to $\mathbb{C}$. We suppose that $f$ is one to one ...
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### radius of convergence of Taylor series, function with branch cuts

Let $f(z)$ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
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### Designing a Power Series with certain $R$

Out of interest, is there a way to design a series with a certain radius of convergence? For example, $R=8$, or is there a way to turn a series for which the Radius of Convergence is known, then ...
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### Radius of convergence of complex power series using Cauchy's integral formula

I have a question as follows. Let $$f(z)=\frac{\sin z}{(z-1-i)^2}$$ and $$a_n=\frac{f^{(n)}(0)}{n!}$$ Determine the radius of convergence of $$\sum_{n=0}^{\infty}a_nz^n$$ In my class we have ...
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### $n$th root of power series when its coefficients are from a field with positive characteristic

Let $k$ be algebraically closed field of characteristic $p>0$. Let's consider a power series $f(x,y)\in k[[x,y]]$. Under what conditions (on $n$, $f$, ...) there exists $g(x,y)\in k[[x,y]]$ such ...
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### Find the fourth Taylor polynomial of f(x)=ln(x+1) at x=1

Let $f(x)=\ln(x+1)$ then (a) find the fourth Taylor polynomial of f at x=1 and (b) use part (a) find the approximate the value of ln(2.2) correct 4 decimal (c) Find an estimate for the error in ...
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### Let $s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$, $c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$ prove that: $c(x)^2-s(x)^2=1$

Let $$s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$$ $$c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$$ prove that $$c(x)^2-s(x)^2=1$$ I know that the following series are representations of the cosh ...
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### Does this matrix series have an answer?

I'm trying to solve this series: $$\displaystyle\sum_{i=0}^{k}A^i B C^{k-i}$$ Where A, B, and C are $N\times N$ symmetric matrices. And $A$ and $C$ have spectral radii smaller than or equal to 1, i....
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### Frobenius Method recurrence relations

Q: By seeking a power series solution to $$2xy′′+(3−x)y′−y = 0$$ about $x=0$ show that there are two linearly independent solutions that have the recurrence relations $$a_{n+1} =\frac{a_n}{2n+3}$$ ...
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### Hadamard's theorem; redefining indexing variable

I have seen in a few proofs the use of Hadamard's theorem to prove convergence of series like this: $\sum_{n\geq 0}z^{n!}$, or $\sum_{n\geq 0}z^{n^2}$ through simply changing the variable of indexing ...
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### What is the radius of convergence of $\sum _{n=0}^{\infty }\left(4+\left(-1\right)^n\right)^nx^n$

$$\sum _{n=0}^{\infty }\left(4+\left(-1\right)^n\right)^nx^n$$ I was asked to find the radius of the convergence, but the Power-Series diverges, so I'm a bit confused. We tried Cauchy-Hadamard and ...
I have the function $f(z) = \frac{3iz-6i}{z-3}$ I need to find a power series $\sum c_n (z-1)^n$ about $z_0 = 1$ I can rewrite $f$ as $\frac{2i-iz}{1-\frac{z}{3}}$, where I'm guessing the ROC ...