Linked Questions

7
votes
1answer
265 views

What functions satisfy $\sum _{n=1} ^{\infty} x_n < \infty \ \Rightarrow \sum _{n=1} ^{\infty} f(x_n) < \infty $ [duplicate]

Could you help me with this problem? What functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy the following implication? $\sum _{n=1} ^{\infty} x_n < \infty \ \Rightarrow \sum _{n=1} ^{\...
-1
votes
2answers
128 views

$\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges [duplicate]

For specific functions and $a_n >0$, we can say that $\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges. I want to show this specifically for the case that $f(x)=sin(x)$. This is what I have thus ...
2
votes
0answers
116 views

$\sum a_n$ is a convergent series of real numbers, is $\sum \frac {a_n}{1+|a_n|}$ also convergent? [duplicate]

If $\sum a_n$ is a convergent series of real numbers, then is $\sum \dfrac {a_n}{1+|a_n|}$ also convergent? I have tried Abel's, Dirichlet's test; nothing's working.
1
vote
1answer
32 views

Convergent sequences transformed into convergent sequences [duplicate]

Problem. Find all the $f: \mathbb{R} \to \mathbb{R}$ (not supposed continuous) such that for every real sequence $(a_n)$ we have : $$\sum a_k \; \text{is convergent} \Longrightarrow \sum f(a_k) \; \...
11
votes
4answers
488 views

$f$ such that $\sum a_n$ converges $\implies \sum f(a_n)$ converges

Let $f:\mathbb R\to \mathbb R$ be such that for every sequence $(a_n)$ of real numbers, $\sum a_n$ converges $\implies \sum f(a_n)$ converges Prove there is some open neighborhood $V$, such ...
8
votes
3answers
3k views

I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$

I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)$. I know that there are other questions that are asking the same thing, but I'm trying to figure this out ...
4
votes
1answer
107 views

Functions on real line which preserves dfferent modes of convergence and preserves divergence of real infinite series

From this question The set of functions which map convergent series to convergent series , it is known that the set of functions on real line which maps convergent series to convergent series is well-...
1
vote
2answers
84 views

Series $a_n$ converges implies series of function $f(a_n)$ converges

Let $f : \mathbb{R} \to \mathbb{R} $ has the following property: For every series $\sum_{n=1}^{\infty} a_n$ converges implies $\sum_{n=1}^{\infty} f(a_n)$ converges. Show that there exist $M>0$ ...
1
vote
1answer
77 views

Comparative status of $\sum a_n$ and $\sum {a_n}^{[2]}$.

Let $(a_n)_n$ be a sequence of real terms. I'm interested in finding a counter-example to the assertion $$\sum\limits_{n=0}^{\infty}a_n\text{ converges }\implies\sum\limits_{n=0}^{\infty}{a_n}^{[2]}...
1
vote
1answer
37 views

Local Linearity and Continuity of $x\sin{1\over x}$

Let $f:\Bbb R\rightarrow \Bbb R\;$ s.t. $f(x) =x\sin{1\over x}$ Q1 : Is the $f$ continuous at $x=0$ ? If so, what is the value of $f(x)$ at $x =0$ ? My Solution : Since $\sin{1\over x}$ trapped in a ...
1
vote
1answer
28 views

Which functions map nonnegative convergent series to convergent series?

Inspired by another question about The set of functions which map convergent series to convergent series (in the domain of the function), I ask the same question but about nonnegative series, which is ...