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 \(|x|<1\)

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TopNewest@Kartik Sharma @Michael Mendrin @Pi Han Goh @Ishan Singh

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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.

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@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"

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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)).

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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).

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video already. I only know as much as you.

I assumed you've watched thisWhether 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.

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@Kartik Sharma I got: \[ \displaystyle \int_{0}^{\infty} \dfrac{\sin(mx)dx}{1-e^{-x}} = \dfrac{1}{2m} + \dfrac{\pi}{2}\coth(\pi m) \]

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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 \( \frac12(\pi \coth(\pi) + 1) \).

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@Pi Han Goh It's not about the sum provided in the beginning, it's about the series given at the end

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