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A Curious Series!

I was dwindling with the Basel series and thinking of ways to prove it. I did this: \[ \displaystyle \sum_{n=0}^{\infty} \dfrac{1}{n^{2}+1} = \int_{0}^{1} \sum_{n=0} ^{\infty} x^{n^{2}}dx \] So, the series is: \[ \displaystyle \sum_{n=0}^{\infty} x^{n^{2}}\] for \( |x| <1 \) Wolfram Alpha gives the result in terms of Jacobi Theta Functions, which I currently fail to comprehend, So, what are your views on it??

And, more generally for the series: \[ \displaystyle \sum_{n=0}^{\infty} x^{n^{\alpha}}\] where, \(\alpha\) is a positive integer or maybe a real, and \(|x|<1\)

Note by Upanshu Gupta
10 months, 4 weeks ago

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I don't know why you want to convert it into an integral (which is wrong by the way). You can simply solve it via digamma functions or residues to get the answer of \( \frac12(\pi \coth(\pi) + 1) \). Pi Han Goh · 10 months, 4 weeks ago

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@Pi Han Goh @Pi Han Goh It's not about the sum provided in the beginning, it's about the series given at the end Upanshu Gupta · 10 months, 4 weeks ago

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@Ishan Singh @Ishan Singh Well I was just calculating \(\zeta(2)\) using the aforesaid method, when I encountered the series. Upanshu Gupta · 10 months, 4 weeks ago

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@Upanshu Gupta Your question is not clear. What do we need to do? It is Jacobi functions, yeah. You want to know about what Jacobi Theta is or other methods to solve the problem?

BTW, you can use Laplace transform too. Kartik Sharma · 10 months, 4 weeks ago

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@Kartik Sharma @Kartik Sharma I got: \[ \displaystyle \int_{0}^{\infty} \dfrac{\sin(mx)dx}{1-e^{-x}} = \dfrac{1}{2m} + \dfrac{\pi}{2}\coth(\pi m) \] Upanshu Gupta · 10 months, 4 weeks ago

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@Upanshu Gupta Yeah. BTW, yeah, I am thinking about the series only for a moment. I too have not yet got any chance to enjoy Theta functions but have heard a lot about them. I would wish to learn about them at the end of my course with Wilfred Kaplan's book(here course means self study of Kaplan). But yeah, I wish to think of these without any theta. Kartik Sharma · 10 months, 4 weeks ago

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@Kartik Sharma @Kartik Sharma Sorry, for mingling the prob like this , but what I want in particular is the summation of that particular series and that if it can be put up in a closed form other than that of jacobi functions.My point is not emphasized on Jacobi functions, but the "series" Upanshu Gupta · 10 months, 4 weeks ago

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@Upanshu Gupta I don't mean to ruin your fun but if there's a simpler way to evaluate \( \displaystyle \sum_n q^{n^2} \) without using Jacobi Theta functions, mathematicians would have known about it already. Pi Han Goh · 10 months, 4 weeks ago

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@Pi Han Goh This is experience.

Ramanujan "guessed" it, why can't we? (This is based on the fact that many people believe that Ramanujan guessed almost every one of his results ans didn't provide results... He should not be regarded as a true mathematician(sarcasm of course)). Kartik Sharma · 10 months, 4 weeks ago

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@Kartik Sharma You made a fair point.

The only thing that I can think of is Rogers-Ramanujan Identities but I doubt that's even relevant to the question at hand.

Yeah, Pierre de Fermat is not a mathematician either (sarcasm). Pi Han Goh · 10 months, 4 weeks ago

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@Pi Han Goh Okay, now the following is random but I just wanted to know more about Skewes' number and more about the shift of inequality sign of \(\pi(x)\) and \(\text{Li}(x)\), can you help? Kartik Sharma · 10 months, 4 weeks ago

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@Kartik Sharma I assumed you've watched this video already. I only know as much as you.

Whether Riemann hypothesis is true or not, the inequality \(\pi(n) < \text{li}(n) \) will fail for a number lower than \(e^{e^{e^{79}}} \) or for a number larger than \(10^{10^{10^{10^3}}} \). Since I'm not verse in Logarithmic Integral, I doubt I can push further than this. Pi Han Goh · 10 months, 4 weeks ago

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