Skip to main content
MathOverflow

Return to Question

Post Closed as "off topic" by Benoît Kloeckner, Andreas Blass, Bill Johnson, Pietro Majer, Felipe Voloch
I add a stronger assumptions.
Source Link
Twi
Twi
  • 1
  • 1

A simple question from mathematical analysis (assumption changed)

Let $\forall n=0,1,2,\dots$, $\alpha_{n}$$\alpha_{n}(x)$ are continuous functions onPOLYNOMIALS in $\mathbb{R}$$x$. Next, let for all $x\neq0$ the power series $$\sum_{n=0}^{\infty}\alpha_{n}(x)t^{n}$$ has positive radius of convergence. Can one say the radius of convergence of the series $$\sum_{n=0}^{\infty}\alpha_{n}(0)t^{n}$$ is also positive?

My quess is NO, but I have no example.

A simple question from mathematical analysis

Let $\forall n=0,1,2,\dots$ $\alpha_{n}$ are continuous functions on $\mathbb{R}$. Next, let for all $x\neq0$ the power series $$\sum_{n=0}^{\infty}\alpha_{n}(x)t^{n}$$ has positive radius of convergence. Can one say the radius of convergence of the series $$\sum_{n=0}^{\infty}\alpha_{n}(0)t^{n}$$ is also positive?

My quess is NO, but I have no example.

A simple question from mathematical analysis (assumption changed)

Let $\forall n=0,1,2,\dots$, $\alpha_{n}(x)$ are POLYNOMIALS in $x$. Next, let for all $x\neq0$ the power series $$\sum_{n=0}^{\infty}\alpha_{n}(x)t^{n}$$ has positive radius of convergence. Can one say the radius of convergence of the series $$\sum_{n=0}^{\infty}\alpha_{n}(0)t^{n}$$ is also positive?

My quess is NO, but I have no example.

Source Link
Twi
Twi
  • 1
  • 1

A simple question from mathematical analysis

Let $\forall n=0,1,2,\dots$ $\alpha_{n}$ are continuous functions on $\mathbb{R}$. Next, let for all $x\neq0$ the power series $$\sum_{n=0}^{\infty}\alpha_{n}(x)t^{n}$$ has positive radius of convergence. Can one say the radius of convergence of the series $$\sum_{n=0}^{\infty}\alpha_{n}(0)t^{n}$$ is also positive?

My quess is NO, but I have no example.