Questions tagged [summation]

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Summation of the series $s_b(p)=\sum_k b^{k^p}$ by a double sum in a sense like Ramanujan-method

From some older context I am re-considering the following variant of the geometric series $$s_b(p)=\sum_{k=1}^\infty b^{k^p}$$ for the convergent cases $0 \lt b \lt 1$ and $0 \lt p$ first. I'm ...
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Sum $\sum_{n\geqslant 1} \frac{(-1)^n \sin(n\theta)}{n^3}$ using complex analysis

The following question appeared on an old complex analysis exam. (The only tools we have learned are simple series for $\sin$, $\cos$, $\exp$ etc., the residue theorem and some tools from real ...
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Simplifying an infinite sum

Question: The sum of $$1-\frac16+\frac16\times\frac14-\frac16\times\frac14\times\frac{5}{18}+\cdots$$ is: A) $\frac23$ B)$\frac{2}{\sqrt3}$ C)$\sqrt\frac23$ D)$\frac{\sqrt3}{2}$ After looking ...
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Computing closed form of $\sum_{k=0}^n b^{\alpha^k}$

Let $$\sum_{k=0}^n b^{\alpha^{k}}$$ be a sum where $b \in \mathbb N_+$ and $\alpha \in \mathbb R, \alpha > 1$. What is its name and how can I calculate its closed form? \begin{align} \sum_{k=0}^n ...
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Replacing $n!$ with Stirling's approximation in $e^x = \sum_n \frac{x^n}{n!}$

I was wondering if there is a closed-form expression for $$\sum_{n=0}^{\infty} \frac{x^n}{e^{-n}n^n},$$ although I expect there is none because Mathematica cannot compute it. However, from Stirling'...
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On geometric series and the logarithmic integral

With this I'm recycling a previous deleted post that contained a serious mistake (in my calculation a geometric series was not convergent). Also was interested in a diffrent question I am asking it ...
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What does $\sum_{n=0}^\infty \sqrt[2^n]{2}-1$ converge into?

I got the following sum from a friend as a challenge: $$\sum_{n=0}^\infty \sqrt[2^n]{2}-1$$ I already know it converges to approximately $1.78183863$, and the full number is irrational doesn't have a ...
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