All Questions
Tagged with algebra-precalculus summation
977
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Example of a specific polynomial
I need four non-trivial polynomials $P(x)$, $Q(y)$, $R(z)$ and $S(w)$ such that $$P(x)Q(y)R(z)S(w)=\sum_{i}a_i b_i c_i d_i x^i y^i z^i w^i+ \sum_{j\neq k\neq l\neq m}e_j f_k g_l h_m x^j y^k z^l w^m $$ ...
- 910
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Multiple Sigma Notation and Expected Value
I've been struggling with expected value calculations that involve multiple or nested sigma notations. For instance, the solution to a problem I was working on is:
$X=\Sigma_{i=1}^{10}X_i\implies E[X]...
- 111
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Summation limits with change of variables
I have the following sum (this is the context of QFT for period finding):
I am having trouble understaning the change in variables from line (1) to line (2).
I understand the sum over m gives ...
- 131
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Double Summation over a Subset of a Cartesian Product
From the "Probability & Statistical Inference, 9th edition" by Hogg, Tannis, Zimmerman, it is stated that one of the properties of the Joint Probability Mass Function of Random Variables ...
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Second-order partial derivative of $S = \sum_{k=0}^{2n}\left(\sum_{i=\max(0,k-n)}^{\min(n,k)}a_{i}a_{k-i}\right)$ [closed]
Given the following finite sum:
$S = \sum_{k=0}^{2n}\left(\sum_{i=\max(0,k-n)}^{\min(n,k)}a_{i}a_{k-i}\right)$
From this summation, I want to calculate explicitly each element $(i,j)$, i.e. the ...
- 85
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Product of two finite sums $\left(\sum_{k=0}^{n}a_k\right) \left(\sum_{k=0}^{n}b_k\right)$
What is the product of the following summation with itself:
$$\left(\sum_{k=2}^{n}k(k-1)a_{k}t^{k-2} \right) \left(\sum_{k=2}^{n}k(k-1)a_{k}t^{k-2}\right) $$
Is the above equal to the double summation ...
- 85
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How to change index in summation
Pease help me understand how they have changed index of summation from r to n here. If we take $$n = r-s$$ how n is changing from -$\infty $ to $\infty$
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Find the value $\sum_{n=a}^b\frac1{\sin (2^{n+3})}$ [closed]
Find the value of: $$\sum_{n=0}^{10}\frac1{\sin (2^{n+3})}$$
I'm stuck on this problem, can someone please help?
- 596
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Double summation with binomial coefficients
Question :
Find the value of the following expression :
$$ \frac{\sum_{i=0}^{2024}\sum_{r=0}^{2024}(-1)^r{2024 \choose r}(2024-r)^i}
{\sum_{r=0}^{2025}(-1)^r\binom{2025}{r}(2025-r)^{2025}} $$
I am not ...
- 37
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125
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Pi/product notation property applications problem
I have recently attempted to simplify this
$$ P(n) = \prod_{v=2}^{n} (2 + \frac{2}{v^2 - 1}) $$
I have reached an answer (which is wrong) through the following steps:
rearranging what is inside the ...
- 15
2
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289
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$\lim_{n\to \infty}(\lim_{x\to 0}( 1+\sum_{k=1}^n\sin^2(kx))^\frac{1}{n^3x^2} )$
$$\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n\sin^2(kx)\right)^\frac{1}{n^3x^2} \right)$$
$$=\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n(k^2x^2)\frac{\sin^2(kx)}{k^2x^2}\...
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2
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53
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Understanding the limit breakdown in a summation problem
Dear fellow members of Math.SE,
I am currently facing a challenge in comprehending a solution to an example problem presented in Donald Knuth's Concrete Mathematics. The problem is as follows:
\begin{...
- 145
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Simplifying Expressions involving Sigma
After reviweing the solutions to a question involving the Biomial Theorem, I arrived at a step, where i was unsure how it occured.
Specifically, i was confused about the logic of:
k=0 -> k=1
n-1 -&...
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115
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How to rewrite $\frac{1}{\sum_{j=1}^{N}\frac{1}{R_j}}$?
This is a very simple question but I don't seem to find a solution online.
I would like to rewrite this sum which appears very often in series and parallel electrical circuits.
$$R_{\text{Eq}} = \frac{...
- 43
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$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$
$$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$$
I solved it by letting the sum be $S$, then adding the sum to itself but taking the terms from last to first and then ...