# Questions tagged [power-series]

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.

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### Monotonicity of quotient of power series

How can it be proved that $$f(x) = \frac{\sum_{n=0}^{\infty} \frac{n}{2 x} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}}{\sum_{n=0}^{\infty} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}}$$ is monotonic? The plot of $f(x)$ ...
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### Is there such a thing as an intermediate value theorem in complex analysis?

I have an exercise I want to solve, but got stuck in the second part due to not having something like the intermediate value theorem for the complex plane. The exercise is: Let $\Omega$ be an open set ...
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### For holomorphic function $f: \Omega \to \mathbb{C}$ with $f^{(k)}(z_0) \in \mathbb{R}$ prove that $f(x) \in \mathbb{R}$ for every $(z_0 - r, z_0 +r)$

I have some difficulties with a question I have come across. The question goes as follows: Let $\Omega$ be an open set with $z_0 \in \Omega \cap \mathbb{R}$. Let $f: \Omega \to \mathbb{C}$ be ...
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### On the naturality of the definition of a deformation?

Let $(A,\mu_0)$ be an associative unital $k$-algebra over a field $k$. A formal deformation of $(A,\mu_0)$ is sometimes defined as a $k[[t]]$-bilinear map $\mu: A[[t]]\times A[[t]]\rightarrow A[[t]]$ ...
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### How to convert first order singular ODE to integral equation?

I have the following ODE: $$v(x)y'=a(x)y^2+b(x)y,~~~y(0)=\varphi,~~v(0)=v(1)=0. \label{eq1}$$ I know that for a similar equation of the form $\varepsilon y'=a(x)y^2+b(x)y~~$...
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I have to show that $$k[[x,y]]/(y^2-x^2) \simeq k[[x,y]]/(y^2-x^3-x^2)$$ where $k[[x,y]]$ is the ring of formal series in $x,y$ over an algebraically closed field $k$. I have the hint that the map $y \... • 4,500 0 votes 0 answers 75 views ###$k[[t]]$-module structure on coordinate ring Let$k$be a field and$k[[t]]$the ring of formal power series over$k$. A text that I am reading says: Consider the coordinate ring$k[x,y,t]/(x^2+txy)$with its canonical$k[[t]]$-module structure.... • 1,653 0 votes 0 answers 41 views ### Expressing$ \sum_{\lambda\in\text{Spec}(\mathbf{X})} \lambda \log \lambda$Say$\text{Spec}(\mathbf{X})$is a set of eigenvalues of a real positive definite matrix$\mathbf{X}$. How do I express $$\sum_{\lambda\in\text{Spec}(\mathbf{X})} \lambda \log \lambda$$ without ... • 525 0 votes 0 answers 63 views ### Series of Analytic Functions is Analytic Let$0 \in \mathbf{N}$. Let$P_m(x): [0,1] \to \mathbf{C}$be bounded analytic functions for every$m\in \mathbf{N}$. Formally, define $$f(x) = \sum_{m\in \mathbf{N}}c_m P_m(x)\overline{P_m}(x),$$ ... -4 votes 1 answer 163 views ### Solution of$ 1 + \frac{x}{2!} + \frac{x^2}{4!} + \frac{x^3}{6!} + \frac{x^4}{8!} + \dots = 0\$ [closed]

So according to the book, the solution is true but is this even possible? The equation is: $$1 + \frac{x}{2!} + \frac{x^2}{4!} + \frac{x^3}{6!} + \frac{x^4}{8!} + \dots = 0 \$$ It asks the solutions, ...