Iterations on my calculator (up to $n=20$) give $x=0.438...$ which is very close to $x=\sin\left(\frac{13\pi}{90}\right)$ although $x$ does not actually converge to it.
Can anyone provide hints as to how I can solve this problem?
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1The solution up to 30 digits is: $x=0.438597975797564612328218966502$ and $\sin(13\pi/90)=0.438371146789077417452734540658$ – asomog Sep 02 '17 at 08:53
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I agree that $x$ doesn't actually converge to $\sin\left(\frac{13\pi}{90} \right )$ (also $\sum_{x=1}^{200}\sin^x\left(\frac{13\pi}{90} \right )\sqrt{x}=0.9989...<1)$. Could $x$ be written as a combination of functions instead of a never-ending decimal? – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Sep 02 '17 at 09:07
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1The series equals $\operatorname{Li}_{-1/2}(x),$ where $\operatorname{Li}_s$ is the polylogarithm function. – md2perpe Sep 02 '17 at 09:30
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Wolfram Alpha only gives an approximation, not a closed form: https://www.wolframalpha.com/input/?i=solve+polylog%5B-1%2F2,x%5D+%3D+1 – md2perpe Sep 02 '17 at 09:39
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If you want a better approximation $\frac{3 \sqrt{9+\sqrt{19}}}{25}\approx 0.4385979307$ – Claude Leibovici Sep 02 '17 at 09:55
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And, for $x=\frac{3 \sqrt{9+\sqrt{19}}}{25}$, the rhs would be $\approx 0.9999997814$ – Claude Leibovici Sep 02 '17 at 10:04
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Very close. I have also discovered that $x$ is approximately $\frac{25}{57}$ although it is not as near as yours. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Sep 02 '17 at 10:06
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@TheSimpliFire. If you want some rational numbers : $\frac{5168}{11783}$ makes the rhs to be $1.00000002104$, $\frac{10361}{23623}$ makes the rhs to be $1.00000000364$ – Claude Leibovici Sep 02 '17 at 11:51
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The power series $$f(x)=\sum_{n=1}^{\infty} \sqrt{n} x^n$$ is convergent in $(-1,1)$ (it can be shown that $f(x)$ is the polylogarithm $\mathrm{Li}_{-1/2}(x)$). It is easy to see that $f$ is continuous, strictly increasing in $[0,1)$. Since $f(0)=0$ and $\lim_{x\to 1^-}f(x)=+\infty$ then by the intermediate value theorem there is a unique $x_0\in (0,1)$ such that $f(x_0)=1$. I would be very surprised if $x_0$ has a closed form (such as $\sin\left(\frac{13\pi}{90}\right)$).
Robert Z
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