Tagged Questions

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Is the sum of reciprocals of all products from $2$ to $n-1$ always $0.5n-1$?

I was looking up riddles for my math classes to work on for the end of the year and found the following riddle. http://mathriddles.williams.edu/?p=129 I followed the advice and started working with ...
18 views

Deriving the identity: $\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

For some reason I am having an extremely hard time finding out how the following expression is derived $$\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$ Is ...
49 views

Is there any closed form expression for the following sum $\sum_{t=1}^T \frac{2^t}{t} = ?$

Is the result upper bounded by $E_i(\ln(2)T)$ ? Edit: where $$E_i(y) = \int_{-\infty}^y \frac{\exp(z)}{z} \mathbb{dz}$$
188 views
+100

Help with proving a statement based on riemann sums?

Suppose we have a reimmen sum with no removed partitions which I call the "total sum". $$\lim_{n\to\infty}\sum_{i=0}^{n}f\left(a+\left(\frac{b-a}{n}\right)i\right)\left(\frac{b-a}{n}\right)$$ And ...
90 views

if $\sum\limits_{i=1}^n{x_i} = 1$, how do you choose the $x_i$'s such that $\sum\limits_{i=1}^n{x_i^2}$ is minimized?

if $\sum\limits_{i=1}^n{x_i} = 1$, how do you choose the $x_i$'s such that $\sum\limits_{i=1}^n{x_i^2}$ is minimized. I have an intuition that each $x_i = \frac{1}{n}$, but I don't know how to prove ...
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What's wrong with my Taylor -Maclaurin- Series? $e^{x^2+x}$

Here's what I have: We know: $$e^x = 1 + x + \frac{1}{2!}x^2+\frac{1}{3!}x^3 +\frac{1}{4!}x^4$$ Now I can calculate the Taylor Series for $e^{x^2+x}$: ...
59 views

Solving a series of equations

I'm writing a piece of code to translate some data, and I keep banging my head against the wall with a one part of the transformation. I'm not as good at math as I ought to be :) Say we have a ...
50 views

How to solve this equation with implicit sum

I want to know how the authors of this arxiv paper (p. 10) solved the equation \begin{align} g\left(\lambda\right) ={}& ...
93 views

Is there any summation method that assigns $\sum_{n=1}^\infty \frac{1}{n} =-\frac{\pi}{2}$

I don't know too much about alternate summation methods, but am interesting to know if any give the sum of the harmonic series to be $$-\frac{\pi}{2}$$
51 views

Why is $\sum_{i=0}^{+\infty} a^{i}i=\frac{a}{(1-a)^{2}}$?

I saw this series in some mathematical proofs but I couldn't find why $\sum_{i=0}^{+\infty} a^{i}i=\frac{a}{(1-a)^{2}}$
111 views

My formula for sum of consecutive squares series?

I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers. For exmaple, this first number in the ...
813 views

Plotting discrete time signals involving sumations in matlab.

Many of the examples I've encountered while looking for an answer are simple functions that do not involve summations. Suppose I have the following function; ...
15 views

Accurate summation of mixed-sign floating-point values

Due to the finite representation, floating-point addition loses significant bits. This is particularly noticeable when there is catastrophic cancellation, such that all the significant bits can ...
95 views
+50

Find a closed form formula for $\sum_{k=2}^n\left(\frac{\sin x}{\sin\frac{x}{k}}\right)^2$ or $\sum_{k=2}^n(\csc\frac{x}{k})^2$?

I meant by "closed form formula" a formulate that doesn't have summation or has very few terms. Maybe there's a better term for this meaning. I found this function that has very interesting property ...
39 views

Evaluate the following trignometric sum

I am interested in the following sum $$\sum_{\text{even } n=-\infty}^{\infty}\left(-\cos^2x\delta_{n,0}+\cos x\left(\frac{1-\cos x}{\sin x}\right)^{|n|}\right).$$ Wolfram alpha returns answer ...
4k views

Sum of combinations of n taken k where k is from n to (n/2)+1

I wonder if there's a formula for obtaining the sum of $n\choose k$'s where $k$ is from $n$ to $\frac{n}{2}+1$. I found out that in odd numbers, it is $2^{n-1}$ (powerset divided by $2$). 1 = 1 3 = ...
34 views

Summing Over Uncountable Index Sets

In answering the question Why do we classify infinities in so many symbols and ideas?, William's answer asserted that summing over an uncountable index set necessarily results in an infinite sum. I am ...
186 views

Numbers of the form $(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$

I'm looking for numbers of the form $$(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$$ where $p_{i}$ are prime numbers, ...
381 views

Sum over fourth power of the sine

I am considering the sum $$A_m = \sum_{j=0}^m \sin^4\left(\frac{j}{m}\cdot\frac{\pi}{2}\right).$$ I think that for $m>1$ it holds $$A_m = \frac{3m+4}{8},$$ but I can't really get to it.
467 views

What are some physical, geometric, or otherwise useful interpretations of divergent series?

I don't understand what ideas such as Abel, Cesàro summation or other types of sum 'regularization' help us describe. What is the practical application to discussing the 'sum' of sequences that are ...
33 views

Summation with arithmetic series

I have doubts how to solve summation if on the top there is something else than $n$ more specificially: $$\sum\limits_{i=1}^{n+1} i=\frac{(n+1)(n+2)}{2}$$ Is my solution correct? I just replace ...
45 views

Develop a model for determining the optimal production schedule in a manufacturing facility

I have to formulate (linearly) the following problem mathematically: What I tried: 1. Variables Let $x_{ijk} = 1$ if, in month k, product i should be made in production line j, where ...
29 views

Matrix Notation! (Linear Algebra)

Suppose that we have a NxM matrix, where N=rows and M=columns. How could I write nicely a ...
21 views

Infinite sum of Hermite polynomials with same order, but different argument

I am looking for any possible simplification of the following sum for positive reals $\alpha,\beta$ and positive integer $n$: $$\sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha)$$ ...
21 views

implementing double $\sum_{m=1}^{p}\sum_{n=1}^{\infty }$ sum in MATLAB

I want to calculated following double sum: $$\sum_{m=1}^{p}\sum_{n=1}^{\infty }(-1)^{n-1}\Delta H.{erfc\frac{2(n-1)u+u\gamma}{2\gamma \sqrt{p-m}}+erfc\frac{2nu-u\gamma}{2\gamma \sqrt{p-m}} }$$ I have ...
21 views

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+500

Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let ...
39 views

Find the sum of $\binom{2007}{0}+\binom{2007}{4}+…+\binom{2007}{2004}$ [duplicate]

Find the sum of $$S=\binom{2007}{0}+\binom{2007}{4}+\binom{2007}{8}+...+\binom{2007}{2004}$$ My work so far: $$(1+1)^n=2^n=\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}$$ ...
78 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
19 views

Laplace pairs - proof of summation transform

I am studying this question for my finals revision and I'm lost on how to start it, can anyone suggest something? Probably pretty simple but I've hit a dead end. Here's the question: If $F_i(t)$ ...
19 views

Identity for the sum of products of Sinc functions

The Sinc function is defined as follows: $$\mathrm{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{ \pi t} & \mathrm{if} \quad t \neq 0, \\ 1 & \mathrm{otherwise.} \end{cases}$$ I want to show the ...
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Is there a simple formula for calculating $\sum_r\binom{n}{r}$, where $1 \leq r \leq n$?

I am aware that the formula for combinations is: $$\binom{n}{r}=\frac{n!}{(n - r)! \:r!}$$ This gives me the numbers I want for combinations of a certain length. However, in my case, I need the ...
56 views

$a_n = b_n -b_{n-1}$ Prove that $\sum_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists

Let $\{b_n\}$ be a sequence Let $a_n = b_n - b_{n-1}$. Prove that $\sum\limits_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists. I am extremely stuck on this homework problem and ...