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

• A simple example: If $\sum x_n$ converges absolutely, then any $f(x) = O(x)$ has the desired property. Mar 6, 2012 at 5:19
• 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.
– AQP
Mar 6, 2012 at 5:21
• Let me ask this: must $f$ be continuous at $0$? Mar 6, 2012 at 5:25
• I misread your question so deleted the comment I posted. Mar 6, 2012 at 5:30
• Jun 21, 2015 at 14:06

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.

• Great answer! Thank you. Sep 30, 2014 at 16:07
• Do you mean, in applying lemma 2, that $f(-x)=-f(x)$? Hard to see how you'd get $f(x)=-x$. Nov 2, 2014 at 14:06
• You're absolutely right, of course. I fixed it. Nov 3, 2014 at 15:48

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 $0$.

• Ok, next question: must $f$ be differentiable at $0$? Mar 6, 2012 at 5:54

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.

• Can you please provide a short proof why $f$ sends convergent series to convergent series? Mar 6, 2012 at 20:16
• I have written the proof. Mar 6, 2012 at 21:04
• 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.
– Did
Mar 6, 2012 at 21:35
• Aside from Didier's point above, are you certain you can split the series? Mar 6, 2012 at 21:42