![Questions Tagged [Summation]](https://docuprint.cloud68.co/uploads/questions-tagged-summation_1200.jpg)
Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
16,061
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Last number in an upwards addition triangle, given the length of the last row.
In the lowest row the numbers 1 to n are written, then rows above consists of the sums of neighboring elements of the row below it (like in Pascal's triangle) until in the highest row only one number ...
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10
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Closed form for $\sum_{k=0}^{n-1} \sin(m\theta_k)(\pi-\theta_k)$
In the process to compute $\int_0^{+\infty} \frac{t^{m-1}}{1+t^{2n}}{\rm d}t$ (see this thread), I got stuck on this sum :
$$S_{m,n} = \sum_{k=0}^{n-1} \sin(m\theta_k)(\pi-\theta_k)$$
where $m$ and $n$...
-1
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0
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30
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Sum to infinity. [closed]
Could someone please explain how this sum is calculated?
$$\sum_{n=0}^{\infty} \frac{a^n e^{(-a)}}{n!} = 1$$
Thanks
Edit:
$$=e^{-a}\sum_{n=0}^{\infty} \frac{a^n }{n!}=e^{-a}e^{a}=1$$
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1
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44
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How to correctly write down a summation used in a simulation
I am unclear on how to mathematically write a summation that is used in my simulation. I am trying to calculate the fluorescence for a slab with ends of -L and +L. There are currently 200 slices ...
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29
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Relating $\sum_{k=1}^N a_k^2 e^{\frac{2\pi i}{N}k}$ to $(\;\sum_{k=1}^N a_k e^{\frac{2\pi i}{N}k}\;)^2$
Consider the following expression
$$
\sum_{k=1}^N a_k^2 e^{\frac{2\pi i}{N}k}\tag{1}
$$
where $i$ is the imaginary number. How may I relate it to the following expression
$$
\left(\sum_{k=1}^N a_k e^{\...
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2
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39
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write the given summation in terms of $x^n$ instead of $x^{3n}$
I have $\sum_{n\geq0}(2n)x^{3n} =0+2x^3+4x^6+6x^9+...$ , but i want to write this summation in terms of $x^n$ instead of $x^{3n}$ .How can i do it ?
I thought that if i can write $n/3$ in place of $n'...
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0
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34
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Sum of binomial coefficients for a specific sum
I am trying to find the eigenvalues of a matrix, and the degeneracy of each eigenvalue is given by the following expression:
\begin{equation}
deg(2l)=4\sum_{\substack{\{0\leqslant 2i,2k \leqslant L^2:\...
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9
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Negative log-likelihood of Gaussian distributions
In a paper about collective outlier detection, I found the following penalized cost formula.
$$ \sum \limits_{t\notin \cup \left[{\tilde{s}}_i+1,{\tilde{e}}_i\right]}\mathcal{C}\left({\mathbf{x}}_t,{\...
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2
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64
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Product of $n$ terms of sequence where the $n^{th}$ term is of the form $(x^{a^n}+1)$
While practicing from a book I found a product in the form $$(x^{a^1}+1)\cdot(x^{a^2}+1)\cdot(x^{a^3}+1)\cdot(x^{a^4}+1)$$ and was immediately curious if I could a formula to solve the product for $n$ ...
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How to switch from continuous to discrete formulation (integral to sum) in a specific case?
in the paper "Economic conditions and the popularity of parties: a survey" Kirchgaessner (1986) transforms a utility function from continuous to discrete. I get the intuition and the meaning,...
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2
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76
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Evaluating $\sum_{k=0}^{\infty}\frac{1}{2k-1}$
$\sum_{k=0}^{\infty}\frac{1}{2k-1}$ is a convergent series.
Is there some way to evaluate $\sum_{k=0}^{\infty}\frac{1}{2k-1}$
This does not look like arithmetic or geometric series to me.
Please help
4
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1
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Infinite symmetrical matrix sum (discrete Lyapunov equation)
I have 2 symmetrical matrices ($A$ and $B$) and I am looking to find the sum $S$:
$$S=A+BAB+B^2AB^2+\ldots$$
Or in summation format:
$$S=\sum_{i=0}^\infty B^iAB^i$$
We know that the absolute magnitude ...
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How to derive the swap rate for a market maker in the given example?
I'm struggling with a section of Financial Mathematics for Actuaries, Second Edition, by Wai-Sum Chan and Yiu-Kuen Tse. Any help would be appreciated! Can't quite seem to figure this one out.
I think ...
2
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1
answer
70
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Closed form of $\sum_{k=1}^\infty\frac{k^nB_k}{k!}$
I developed the following:
Consider
$$\frac{t}{e^t-1}=\sum_{k=0}^\infty\frac{B_k}{k!}t^k,$$
where $B_k$ are the Bernoulli numbers, then
$$\frac{e^t}{e^{e^t}-1}=\sum_{k=0}^\infty\frac{B_k}{k!}e^{tk}=\...
-5
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1
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Prove by using binomial theorem on $(x-1)^n$ the following [closed]
Prove by using binomial theorem on $(x-1)^n$ the following:
$$\sum_{k=0}^{n-1}(-1)^k(n-k)C^k_n=0$$
