Questions tagged [summation]
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Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
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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 ... summation pattern-recognition- 1
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 $n$... summation- 4,654
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: $$=e^{-a}\sum_{n=0}^{\infty} \frac{a^n }{n!}=e^{-a}e^{a}=1$$ sequences-and-series summation- 53
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 ... summation- 47
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 $$ \left(\sum_{k=1}^N a_k e^{\... algebra-precalculus summation fourier-transform sums-of-squares- 3,418
write the given summation in terms of $x^n$ instead of $x^{3n}$
I have $\sum_{n\geq0}(2n)x^{3n} =0+2x^3+4x^6+6x^9+...$ , but i want to write this summation in terms of $x^n$ instead of $x^{3n}$ .How can i do it ? I thought that if i can write $n/3$ in place of $n'... sequences-and-series algebra-precalculus summationSum of binomial coefficients for a specific sum
I am trying to find the eigenvalues of a matrix, and the degeneracy of each eigenvalue is given by the following expression: \begin{equation} deg(2l)=4\sum_{\substack{\{0\leqslant 2i,2k \leqslant L^2:\... combinatorics summation binomial-coefficients binomial-theorem- 11
Negative log-likelihood of Gaussian distributions
In a paper about collective outlier detection, I found the following penalized cost formula. $$ \sum \limits_{t\notin \cup \left[{\tilde{s}}_i+1,{\tilde{e}}_i\right]}\mathcal{C}\left({\mathbf{x}}_t,{\... calculus sequences-and-series summation log-likelihood- 1,368
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 $n$ ... sequences-and-series algebra-precalculus summation generating-functions products- 23
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,... integration discrete-mathematics summation riemann-sum- 1
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 algebra-precalculus summation- 810
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 ... matrices summation geometric-series- 679
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 ... summation finance actuarial-science- 137
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 $$\frac{e^t}{e^{e^t}-1}=\sum_{k=0}^\infty\frac{B_k}{k!}e^{tk}=\... summation bernoulli-numbers- 9,883
Prove by using binomial theorem on $(x-1)^n$ the following [closed]
Prove by using binomial theorem on $(x-1)^n$ the following: $$\sum_{k=0}^{n-1}(-1)^k(n-k)C^k_n=0$$ summation binomial-theorem- 1
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