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This is an exercise from Problems from the Book by Andreescu and Dospinescu. When it was posted on AoPS a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone here can enlighten me.

Problem: Prove that the function $f : [0, 1) \to \mathbb{R}$ defined by

$\displaystyle f(x) = \log_2 (1 - x) + x + x^2 + x^4 + x^8 + ...$

is bounded.

A preliminary observation is that $f$ satisfies $f(x^2) = f(x) + \log_2 (1 + x) - x$. I played around with using this functional equation for awhile, but couldn't quite make it work.

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    $\begingroup$ if you differentiate your functional equation you get that limx→1f′(X)lim_{x\to 1} f'(X) exists and is finite. doesn't that do it for you? $\endgroup$ Aug 3, 2010 at 0:46
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    $\begingroup$ Hmm. Maybe. If you wrote that up together with a proof that f' actually exists, I'll accept it. That seems too easy, somehow. $\endgroup$ Aug 3, 2010 at 1:11
  • $\begingroup$ A lot of solutions people claim are "from the book" are short and sweet :) $\endgroup$ Aug 3, 2010 at 1:23
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    $\begingroup$ Plotting the function in Mathematica seems to indicate otherwise... $\endgroup$ Aug 3, 2010 at 1:56
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    $\begingroup$ @Danny: it is only being claimed that the problems, not their solutions, are from the Book! The authors, by their own admission, don't know how to solve some of the exercises... $\endgroup$ Aug 3, 2010 at 2:02

4 Answers 4

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OK, a second trick is needed (but it actually finishes the problem). It is nice and simple enough that it's probably what the authors intended by a "Book" solution.

Let $f(x) = x \log(2) - \log(1+x)$. We want to show that $S(x) = f(x) + f(x^2) + f(x^4) + \dots$ is bounded. Because $f(0)=f(1)=0$ and $f$ is differentiable, we can find a constant $A$ such that $|f(x)| \leq Ax(1-x) = Ax - Ax^2$. The sum of this bound over the powers $x^{2^k}$ is telescopic.

Notice that the role of $\log(2)$ was to ensure that $f(1)=0$.

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    $\begingroup$ Very nice! (and extra characters to satisfy the silly 15 character limit) $\endgroup$ Aug 3, 2010 at 19:21
  • $\begingroup$ "Because f(0) = f(1) = 0 and f is differentiable, we can find a constant A..." ??? could you clarify? I'm missing something, is that the extreme value theorem? en.wikipedia.org/wiki/Extreme_value_theorem $\endgroup$
    – Jason S
    Aug 3, 2010 at 22:25
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    $\begingroup$ g(x) = f(x)/(x(1-x)) is defined on (0,1) and extends to a continous function on [0,1]. The extension exists because the limits of g(x) needed at the endpoints are f'(0) and f'(1). So differentiability of f(x) at 0 and 1 implies continuity of g(x), and for A just take any upper bound on the values of |g(x)| in [0,1]. $\endgroup$
    – T..
    Aug 3, 2010 at 22:39
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Starting from (the natural logarithm of) $(1-x)^{-1} = (1+x)(1+x^2)(1+x^4) \dots$, it becomes clearer where the $\log(2)$ factor comes from.

One has to show that $\Sigma (x^{2^k} - C\log(1 + x^{2^k}))$ is bounded sum of positive terms. The sum of the first $n$ terms approaches $n - Cn\log(2)$ as $x \to 1-$, so we need $C = 1/\log(2)$ if there is to be boundedness.

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    $\begingroup$ Nice observation. Can you finish the argument from here? $\endgroup$ Aug 3, 2010 at 3:07
  • $\begingroup$ Yes and no. I posted a completion of the proof (see other answer) but it requires an additional, independent observation. $\endgroup$
    – T..
    Aug 3, 2010 at 19:04
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If anyone's interested, here's another approach.

First, observe that $$ \log{2} = \int_{2^r}^{2^{r+1}} \frac{1}{x}\;dx \le \sum_{k=2^r}^{2^{r+1}-1}\frac{1}{k} \le \int_{2^r-1}^{2^{r+1}-1} \frac{1}{x}\;dx = \log{2} + \log\left(1+\frac{1}{2^{r+1}-2}\right) $$ for $r\ge1$, so using $\log(1+x) \le x$ (for $x>-1$) and $2^{r+1} - 2 \ge 2^r$, we get $$ \log{2} \le \sum_{k=2^r}^{2^{r+1}-1} \frac{1}{k} \le \log{2} + \frac{1}{2^r} $$ for all $r\ge0$ (check directly for $r=0$).

Thus for $x\in[0,1)$, we have $$\begin{align*} \lvert f(x)\log{2}\rvert = \lvert\sum_{r\ge0} x^{2^r}\log{2} - \sum_{n\ge1} \frac{x^n}{n}\rvert &= \lvert\sum_{r\ge0} x^{2^r}(\log{2} - \sum_{k=2^r}^{2^{r+1}-1}\frac{1}{k}) + \sum_{r\ge0}\sum_{k=2^r}^{2^{r+1}-1}\frac{x^{2^r} - x^k}{k} \rvert \\ &\le \sum_{r\ge0}\frac{x^{2^r}}{2^r} + \sum_{r\ge0}\sum_{k=2^r}^{2^{r+1}-1}\frac{x^{2^r} - x^k}{k} \\ &< 2 + \sum_{r\ge0}x^{2^r}(1-x)\sum_{k=2^r}^{2^{r+1}-1}\frac{1+x+\cdots+x^{k-2^r-1}}{k} \\ &\le 2 + (1-x)\sum_{r\ge0}x^{2^r}\sum_{k=2^r}^{2^{r+1}-1}\frac{k-2^r}{k} \\ &\le 2 + (1-x)\sum_{r\ge0}x^{2^r}(2^r\cdot 1) \\ &\le 2 + (1-x)(x + 2\sum_{r\ge1}\sum_{k=2^{r-1}+1}^{2^r}x^k) \\ &= 2 + x(1-x) + 2(1-x)\sum_{k\ge2} x^k \\ &= 2 + x + x^2, \end{align*}$$ so we're done.

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Solution using minimal calculus: Let $y=1-x$ and $z=\lfloor{\log_2(\frac{1}{1-x})}\rfloor$.

Lemma $1$:

$x^{2^{n}} \geq (1-2^{n-1}y),\; \forall n \in \mathbb{N^+}$.

Proof:

We proceed by induction. For the base case $n=1$, we have $1 \geq 1-y$. Suppose this is true for $n$. Then, assuming $1-2^{n}y \geq 0$, $x^{2^{n+1}}=\left(x^{2^{n}}\right)^{2}$ $\geq \left(1-2^{n}y\right)^{2} = 1-2^{n+1}y+2^{2n}y^{2} \geq 1-2^{n+1}y$. If $1-2^{n}y < 0$, then this is obviously true.

We have $$\log_2(1-x)+\sum_{i=1}^\infty x^{2^{i}} \\= \sum_{i=1}^\infty x^{2^{i}}-\log_2\left(\dfrac{1}{1-x}\right) \\\geq \sum_{i=0}^z (1-2^{i}y)-\log_2(1-x)\\= \left\lfloor{\log_2\left(\frac{1}{1-x}\right)}\right\rfloor+1-\log_2\left(\frac{1}{1-x}\right)-\left(2^{z+1}-1\right)y \geq -2.$$

It is also well known that $x^{2^{z+1}} \leq \frac{1}{e}$, so $$\sum_{i=1}^\infty x^{2^{i}}+\log_2(1-x) \\= \sum_{i=1}^\infty x^{2^{i}}-\log_2\left(\dfrac{1}{1-x}\right) \\\leq \sum_{i=1}^z x^{2^{i}}+\sum_{i=1}^\infty x^{i2^{z+1}}-\log_2\left(\dfrac{1}{1-x}\right) \\\leq \left\lfloor {\log_2\left(\dfrac{1}{1-x}\right)}\right\rfloor-\log_2(\dfrac{1}{1-x})+\dfrac{1}{e-1} \\\leq \dfrac{1}{e-1}$$

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