Answer by PseudoNeo for The set of functions which map convergent series to...
I'm quite late on this one, but I think the result is nice enough to be included here.Definition A function $f : \mathbb R \to \mathbb R$ is said to be convergence-preserving (hereafter CP) if $\sum...
View ArticleAnswer by Julián Aguirre for The set of functions which map convergent series...
Answer to the next question: no.Let $f\colon\mathbb{R}\to\mathbb{R}$ be defined by$$f(x)=\begin{cases}n\,x & \text{if } x=2^{-n}, n\in\mathbb{N},\\x & \text{otherwise.}\end{cases}$$Then...
View ArticleAnswer by Patrick for The set of functions which map convergent series to...
If $f$ is not continuous at $0$, then we can find a sequence $x_n$ that converges to $0$ but $f(x_n)$ doesn't converge to $0$. First get a subsequence $y_n$ of $x_n$ with $|f( y_n)| > r$ for some...
View ArticleThe set of functions which map convergent series to convergent series
Suppose $f$ is some real function with the above property, i.e.if $\sum\limits_{n = 0}^\infty {x_n}$ converges, then $\sum\limits_{n = 0}^\infty {f(x_n)}$ also converges. My question is: can anything...
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