# Tagged Questions

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### Closed form expression for the constrained least squares problem

$\underset{A}{\min} \hspace{15mm} \sum_{k=1}^{N-1} ||\textbf{x}_{k+1}-A\textbf{x}_{k}||^2_2$ $s.t \hspace{15mm} W.A = 0$ where (.) is the element by element product and $W$, $\textbf{x}_{i}$ are ...
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### Prove a vector in $\ell^2(\mathbb{Z})$ is zero

Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$ That is, every elements of the vector is a series with the other terms in $\vec{c}$. ...
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### Summation for $\sum\limits^5_{i=2}\:\left(3i\:-\:5\right)$

I know that the closed form of $\sum\limits^n_{k=1}\:k=\frac{n(n+1)}{2}$ But I'm not sure what the closed form for $\sum\limits^5_{i=2}\:\left(3i\:-\:5\right)$ would be. Any push in the right ...
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### Simple expression for $\sum_{k=1}^{n-1}\:\frac{1}{k\left(k+1\right)}$

I know that $\:\:\frac{1}{k\left(k+1\right)}\:\:\:\:=\:\frac{1}{k}\:-\:\frac{1}{k+1}\:$ And that $\sum_{k=1}^{n-1}\:k$ $= \frac{n(n-1)}{2}$ But I'm not completely sure how to turn ...
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### About the sums $\sum_{n=1}^\infty x^{n^2}$ and $\sum_{n=1}^\infty \frac{x^n}{1+x^{2n}}$

Despite all my efforts trying to crack these, i haven't been able to do so. An approach that i've tried gives me somewhat of an asymptotic approximation, but still fails to produce the values near ...
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### Summation of series with binomial coefficients

The value of $$\sum {n\choose n-r} (n-r) \sin(r\cdot \pi/n)$$ where $r\in (0 ..,n)$ is equal to? I think the question can be solved by writing the series in reverse order but I am not able to solve ...
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### Differentiate a geometric sum and show that it is less than an equation

The question: a) Differentiate both sides of the geometric series with respect to $r$: $$~~\displaystyle\sum_{i=0}^nr^i=\frac{1-r^{n+1}}{1-r}$$ b) Use the result in part (a) to show that (Assume ...
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### Finding roots of an equation wich involves floor function

I'm trying to solve this equation $$\left \lfloor{x +\frac{1}{100}}\right \rfloor + \left \lfloor{x +\frac{2}{100}}\right \rfloor + ... + \left \lfloor{x +\frac{223}{100}}\right \rfloor = 521$$ I ...
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### What $+1+1+\cdots$ really equals

$1+1+1+\cdots$ is clearly a divergent series, so you'd say that it tends towards infinity? Through analytic continuation of the zetafunction the value $-1/2$ could be assigned the sum, right? But if ...
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### Closed-form Solution to a Sum

I have some math questions for a programming course where it says to provide closed-form solutions for a list of sums. I've never taken an algorithms course, so I'm not quite sure what I'm doing. I ...
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### Factorial Summation Definition

A while back I found the series $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$ while messing around in Algebra class (specifically when $n$ is any natural number and $x$ is any real number) I ...
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### how can $((\frac1N \sum z_i )- I )^2 = \frac1{N^2}(\sum (z_i - I)^2 )$?

$$\left(\left(\frac1N \sum z_i \right)- I \right)^2 = \frac1{N^2}\left(\sum (z_i - I)^2 \right)$$ How does this work? Or is there an error? I thought that you could not pull the sum out of the ...
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### Summation from Right to Left changes Accuracy?

I was looking at The accuracy from left to right and that from right to left of the floating point arithmetic sums which was asking for the accuracy of $$\sum_{k=1}^n\frac{1}{k^2}$$ from right to ...
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### Does $\sum_{i=1}^n \frac{1}{i^2}=O(\ln(n))$?

I was looking at Why $\sum_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n))$?. And there it was proved that $$\sum_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n))$$ My question is that does this also stand for ...
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### What do $\{ceps_q\}_{q=0}^Q$ and $\{a_q\}_{q=1}^p$ mean?

As a programmer who hasn't had any higher mathematical training, I sometimes find mathematical equations described in books or online that I'd like to implement in my programs, but they have symbols ...
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### Differentiating $- \sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha}\int_0^E\frac{1}{4\pi t}\exp({\omega^2 t - \frac{|x - n - y|^2}{4t^2}})dt$ wrt $x$?

I have a formula for the Ewald method which can be used to speed up computations when working with periodic Green's functions. I will need to take the derivative of the function $G(x, y)$ with respect ...
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