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the graph of the function $$f(x)=\frac{1}{1+x}+\frac{2x}{1+x^2}+\cdots+\frac{2^{n-1}x^{2^{n-1}-1}}{1+x^{2^{n-1}}}+\cdots$$ where $x \in (-1,1)$ is symmetric about the point---

I tried to integrate the function and could not proceed to find centre of symmetry.I tried it in desmos and i got the following curve.Please help me out.Thanks

JMP
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Navin
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1 Answers1

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Surprisingly $f(x)$ is a very simple function as shown below. To get it, we start from :

$F_0(x)=(1+x)$

$F_1(x)=(1+x)(1+x^2)=(1+x)+x^2(1+x)=1+x+x^2+x^3$

$\begin{cases} F_2(x)=(1+x)(1+x^2)(1+x^4)=(1+x^2+x^3)+x^4(1+x^2+x^3) \\ F_2(x)=1+x^2+x^3+x^4+x^5+x^6+x^7 \end{cases}$

and so on ...

$\begin{cases} F_n(x)=(1+x)(1+x^2)...(1+x^{(2^n)}) \\ F_n(x)=1+x+x^2+...+x^{(2^{n+1}-1)} \end{cases}$ $$F_n(x)=\displaystyle\prod_{k=0}^{n}(1+x^{(2^k)})=\displaystyle\sum_{k=0}^{2^{n+1}-1}x^k=\frac{1-x^{(2^{n+1})}}{1-x}$$ $$\displaystyle\prod_{k=0}^{\infty}(1+x^{(2^k)})=\frac{1}{1-x}$$ $$\ln\left(\displaystyle\prod_{k=0}^{\infty}(1+x^{(2^k)})\right) =\displaystyle\sum_{k=0}^{\infty}\ln(1+x^{(2^k)})=-\ln(1-x)$$ The differentiation leads to : $$\displaystyle\sum_{k=0}^{\infty}\frac{2^kx^{(2^k)-1} }{1+x^{(2^k)}}=\frac{1}{1-x}$$ We recognize $f(x)$ : $$f(x)=\frac{1}{1-x}$$ All above is valid for $-1<x<1$. So, $f(x)$ is an arc of hyperbola.

Obviously, the symmetry is with respect to the straight line $y=1-x$ with center $(x=0\:,\:y=1)$.

enter image description here

This is in good agreement with your own graph (magnified) :

enter image description here

JJacquelin
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  • Actuallythe graph is not symmetric over $(-1,1)$ as one arm goes to infinity, while the other arm ends at $-1$. The graph of the restriction of $f$ to $(-1,\frac12)$ is symmetric, though. – celtschk Dec 04 '16 at 07:43
  • @cellschk : I agree. In fact, I mean the symmetry of the whole branch of the hyperbola, including the extension of $f$ to $-\infty$ as the dotted line on the figure. Thank you for the clarification. – JJacquelin Dec 04 '16 at 08:10
  • To the Editors, about the edition suggested : The correction proposed was justified, but not sufficient. I corrected and re-edited all the beginning. This doesn't change the end of the answer. Thank you for pointing out the mess in the beginning. – JJacquelin Dec 04 '16 at 08:21