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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 interesting be said regarding the behavior of such a function close to $0$, other than the fact that $f(0)=0$?

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    $\begingroup$ A simple example: If $\sum x_n$ converges absolutely, then any $f(x) = O(x)$ has the desired property. $\endgroup$ Commented Mar 6, 2012 at 5:19
  • $\begingroup$ All I can think is that , if the space you're in is complete, uniformly-continuous functions take Cauchy sequences to Cauchy sequences. This allows a unique continuous extension for functions defined in dense subsets.And continuous alone is not enough for this last, as f(x)=1/(x-2^{1/2}) is an example. $\endgroup$
    – AQP
    Commented Mar 6, 2012 at 5:21
  • $\begingroup$ Let me ask this: must $f$ be continuous at $0$? $\endgroup$
    – the_fox
    Commented Mar 6, 2012 at 5:25
  • $\begingroup$ I misread your question so deleted the comment I posted. $\endgroup$
    – Patrick
    Commented Mar 6, 2012 at 5:30
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    $\begingroup$ See also $f$ such that $\sum a_n$ converges $\implies \sum f(a_n)$ converges $\endgroup$ Commented Jun 21, 2015 at 14:06

3 Answers 3

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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 $\sum f(a_n)$ converges for every convergent series $\sum a_n$.

Theorem (Wildenberg): The CP functions are exactly the ones which are linear on some neighbourhood of $0$.

Proof (Smith): Clearly, whether $f$ is CP only depends on the restriction of $f$ on an arbitrary small neighbourhood of $0$. Since the linear functions are CP, the condition is clearly sufficient. Let's prove that it is also necessary.

We will prove two preliminary results.

Lemma 1: $f$ CP $\Rightarrow$ $f$ continuous at $0$.

Proof: Let's suppose that $f$ isn't continuous at 0. This implies that there exists a sequence $\epsilon_n \to 0$ and a positive real $\eta > 0$ such that $\forall n, |f(\epsilon_n)| \geq \eta$. But it is easy to extract a subsequence $\epsilon_{\phi(n)}$ such that $\sum \epsilon_{\phi(n)}$ converges (take $\phi$ such that $\epsilon_{\phi(n)} \leq 2^{-n}$, for instance). For such a subsequence, we still have that $|f(\epsilon_{\phi(n)})| \geq \eta$. This prevents $\sum f(\epsilon_{\phi(n)})$ to converge and, thus, $f$ to be CP, a contradiction.

Lemma 2: The function $(x, y) \mapsto f(x+y) + f(-x) + f(-y)$ vanishes on some neighbourhood of $0$.

Proof: If it didn't, one would be able to find sequences $x_n \to 0$ and $y_n \to 0$ s.t. $\forall n, f(x_n + y_n) + f(-x_n) + f(-y_n) \neq 0$. Up to some extraction, we can assume that $\delta_n = f(x_n + y_n) + f(-x_n) + f(-y_n)$ always has the same sign (let's say $\delta_n > 0$, for the sake of simplicity.)

Consider now the series $$\begin{array}{l@{}l} (x_0 + y_0) &+ (-x_0) + (-y_0) + \cdots + (x_0 + y_0) + (-x_0) + (-y_0)\\ & +(x_1 + y_1) + (-x_1) + (-y_1) + \cdots + (x_1 + y_1) + (-x_1) + (-y_1)\\ &+\cdots\\ &+(x_n + y_n) + (-x_n) + (-y_n) + \cdots + (x_n + y_n) + (-x_n) + (-y_n)+\cdots, \end{array}$$ where every triplet of termes $(x_i+y_i) + (-x_i) + (-y_i)$ is repeated $M_i > 0$ times, for some integer $M_i > 0$.

Because $x_n \to 0$ and $y_n \to 0$ and the three terms $x_i + y_i, -x_i, -y_i$ add to 0, it is easy to see that this series is convergent, regardless of the choice of the $M_i$'s.

On the other hand, if we choose $M_i \geq \delta_i^{-1}$, the image of our series by $f$ is $$\begin{array}{l@{}l} f(x_0 + y_0) &+ f(-x_0) + f(-y_0) + \cdots + f(x_0 + y_0) + f(-x_0) + f(-y_0)\\ & +f(x_1 + y_1) + f(-x_1) + f(-y_1) + \cdots + f(x_1 + y_1) + f(-x_1) + f(-y_1)\\ &+\cdots\\ &+f(x_n + y_n) + f(-x_n) + f(-y_n) + \cdots + f(x_n + y_n) + f(-x_n) + f(-y_n)+\cdots, \end{array}$$ which diverges, for every line adds to $M_i \delta_i > 1$. Again, this in direct contradiction with the CPness of $f$.

If we apply the result of lemma 2 with $y = 0$, we get that $f(-x) = -f(x)$. So we can rewrite lemma 2 in the following way: $\exists \eta > 0 : \forall x, y \in (-\eta, \eta), f(x+y) = f(x) + f(y)$.

This property and the continuity at 0 imply first the continuity on the whole of $(-\eta, \eta)$ and it is then not hard to adapt the classical proof to show that $f$ is linear on $(-\eta, \eta)$. Q.E.D.

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  • $\begingroup$ Great answer! Thank you. $\endgroup$
    – the_fox
    Commented Sep 30, 2014 at 16:07
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    $\begingroup$ Do you mean, in applying lemma 2, that $f(-x)=-f(x)$? Hard to see how you'd get $f(x)=-x$. $\endgroup$ Commented Nov 2, 2014 at 14:06
  • $\begingroup$ You're absolutely right, of course. I fixed it. $\endgroup$
    – PseudoNeo
    Commented Nov 3, 2014 at 15:48
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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 $r>0$. Next choose some subsequence $z_n$ of $y_n$ so that $\sum z_n$ converges. However the series $\sum f(z_n)$ diverges and it follows that $f$ is continuous at $0$.

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    $\begingroup$ Ok, next question: must $f$ be differentiable at $0$? $\endgroup$
    – the_fox
    Commented Mar 6, 2012 at 5:54
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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 $\lim_{x\to0}f(x)=f(0)=0$, $f$ transforms convergent series in convergent series, but $f(x)/x$ is not bounded in any open set containing $0$. In particular $f$ is not differentiable at $x=0$. This example can be modified to make $f$ continuous.

Proof.

Let $\sum_{k=1}^\infty x_k$ be a convergent series. Let $I=\{k\in\mathbb{N}:x_k=2^{-n}\text{ for some }n\in\mathbb{N}\}$. For each $k\in I$ let $n_k\in\mathbb{N}$ be such that $x_k=2^{-n_k}$. Then $$ \sum_{k=1}^\infty f(x_k)=\sum_{k\in I} n_k\,2^{-n_k}+\sum_{n\not\in I} x_n. $$ The series $\sum_{k\in I} n_k\,2^{-n_k}$ is convergent. It is enough to show that also $\sum_{n\not\in I} x_n$ is convergent. This follows from the equality $$ \sum_{n=1}^\infty x_n=\sum_{n\in I} x_n+\sum_{n\not\in I} x_n $$ and the fact that $\sum_{n=1}^\infty x_n$ is convergent and $\sum_{k\in I} x_n$ absolutely convergent.

The proof is wrong. $\sum_{k\in I} x_n$ may be divergent. Consider the series $$ \frac12-\frac12+\frac14-\frac14+\frac14-\frac14+\frac18-\frac18+\frac18-\frac18+\frac18-\frac18+\frac18-\frac18+\dots $$ It is convergent, since its partial sums are $$ \frac12,0,\frac14,0,\frac14,0,\frac18,0,\frac18,0,\frac18,0,\frac18,0,\dots $$ The transformed series is $$ \frac12-\frac12+\frac24-\frac14+\frac24-\frac14+\frac38-\frac18+\frac38-\frac18+\frac38-\frac18+\frac38-\frac18+\dots $$ whose partial sums are $$ \frac12,0,\frac12,\frac14,\frac34,\frac12,\frac78,\frac34,\frac98,1,\frac{11}8,\frac54,\dots $$ which grow without bound.


On the other hand, $f(x)=O(x)$, the condition in Antonio Vargas' comment, is not enough when one considers series of arbitrary sign. Let $$ f(x)=\begin{cases} x\cos\dfrac{\pi}{x} & \text{if } x\ne0,\\ 0 & \text{if } x=0, \end{cases} \quad\text{so that }|f(x)|\le|x|. $$ Let $x_n=\dfrac{(-1)^n}{n}$. Then $\sum_{n=1}^\infty x_n$ converges, but $$ \sum_{n=1}^\infty f(x_n)=\sum_{n=1}^\infty\frac1n $$ diverges.

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  • $\begingroup$ Can you please provide a short proof why $f$ sends convergent series to convergent series? $\endgroup$
    – the_fox
    Commented Mar 6, 2012 at 20:16
  • $\begingroup$ I have written the proof. $\endgroup$ Commented Mar 6, 2012 at 21:04
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    $\begingroup$ Not sure $\sum\limits_{k\in I}n_k/2^{n_k}$ always converges. Consider $(x_i)_{i\geqslant1}$ such that every $x_i$ is a negative power of $2$ and assume that $x_i=1/2^k$ roughly $2^k/k^2$ times. Then $\sum\limits_{i\geqslant1}x_i\approx\sum\limits_{k\geqslant1}1/k^2$ converges but $\sum\limits_{i\geqslant1}f(x_i)\approx\sum\limits_{k\geqslant1}1/k$ diverges. $\endgroup$
    – Did
    Commented Mar 6, 2012 at 21:35
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    $\begingroup$ Aside from Didier's point above, are you certain you can split the series? $\endgroup$
    – the_fox
    Commented Mar 6, 2012 at 21:42

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