Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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6
votes
3answers
36 views

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 ...
2
votes
1answer
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 ...
1
vote
2answers
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 $$E_i(y) = \int_{-\infty}^y \frac{\exp(z)}{z} \mathbb{dz}$$
2
votes
2answers
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 ...
0
votes
5answers
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 ...
0
votes
2answers
47 views

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 $e^{x^2+x}$: ...
0
votes
1answer
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 ...
2
votes
1answer
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) ={}& ...
0
votes
1answer
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 $$-\frac{\pi}{2}$$
1
vote
3answers
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 $\sum_{i=0}^{+\infty} a^{i}i=\frac{a}{(1-a)^{2}}$
5
votes
3answers
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 ...
3
votes
1answer
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; ...
1
vote
1answer
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 ...
7
votes
0answers
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 ...
2
votes
2answers
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 ...
3
votes
1answer
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 = ...
2
votes
1answer
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 ...
9
votes
2answers
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, ...
4
votes
1answer
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.
6
votes
4answers
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 ...
2
votes
1answer
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 ...
1
vote
0answers
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 ...
0
votes
1answer
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 ...
0
votes
0answers
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 $n$: $$ \sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha) $$ ...
0
votes
1answer
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 ...
0
votes
1answer
21 views

Can somone explain this specific Summary Notation to me?

Been re-teaching myself all the math I slept through in high school. Really enjoying the challenge, but I hit a wall last night. The workbook I have has the following notation: $$\prod_{i=0}^4 7(i + ...
0
votes
1answer
89 views

Proving the closed form of a generating function of the sum of n lucas numbers is equal to the n+2th lucas number

1760887     I've been working on this homework problem for a while now and can't seem to solve it. Let $L_n = L_{n-1} + L_{n-2}$ for $n\ge 2$ where $L_0 = 2$ and $L_1 = 1$ $M_n = 1 + ...
0
votes
2answers
40 views

How to solve this series :$\sum_{k=0}^{\frac{n}{2}}n-k$

I tried to solve this series as follows ; $\sum_{k=0}^{\frac{n}{2}}n-k$ : $ =(\frac{n}{2}+n)+(\frac{n}{2}+1+n-1)+(\frac{n}{2}+2+n-2)+...+(\frac{n}{2}+k+n-k) = ...
1
vote
2answers
43 views

Alternating series of compositions of triangular numbers

I'm modeling a process which involves a subset $S$ of a large number $n_A$ of objects - call them balls. Each time I add a ball to $S$, it may dislodge another ball with probability proportional to ...
0
votes
1answer
17 views

Formulae for $\sum_{k=0}^{\left\lfloor n/2\right\rfloor }\left(\begin{array}{c} n\\ 2k \end{array}\right)(-1)^{k}a^{n-2k}b^{2k}$?

Given a complex number $z=a+bi$, its $n$th power can be written in closed form as $$(a+bi)^n=\sum_{k=0}^{\left\lfloor n/2\right\rfloor }\left(\begin{array}{c} n\\ 2k ...
5
votes
0answers
199 views
+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 ...
4
votes
0answers
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: $$(1+1)^n=2^n=\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}$$ ...
4
votes
0answers
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 ...
0
votes
1answer
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 $F_i(t)$ ...
0
votes
1answer
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 ...
5
votes
1answer
74 views

Combinatorics problem that deals with trigonometric functions

If $m$ and $p$ are positive integers and $m \geq p$, then show that $${m \choose 0}+{m \choose p}+{m \choose 2p}+{m \choose 3p}+\cdots$$ has value $${2^m \over p}\left(1+\sum_{k=1}^{\left ...
3
votes
3answers
17k views

Finding the Moment Generating function of a Binomial Distribution

Suppose $X$ has a $Binomial(n,p)$ distribution. Then its moment generating function is $$ M(t) = \sum_{x=0}^x {n \choose x}p^x(1-p)^{n-x} \\ =\sum_{x=0}^{n} {n \choose x}(pe^t)^x(1-p)^{n-x}\\ ...
1
vote
2answers
34 views

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 ...
1
vote
1answer
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 ...
0
votes
1answer
25 views

how to solve this nested summation? [on hold]

I would like to know if there is some formula for solving this type of summations: I can solve it by analytically playing with the inner summation, but I do not know if what am I doing is right. ...
0
votes
1answer
30 views

Can someone draw a plot for this function?

Can someone draw a plot for this function? $ f(x) = \begin{cases} \sum_{i=2}^{x}\left(\frac{\prod_{k=1}^{i-1}\left(2k-1\right)\,\cdot\,-\left(-\frac{1}{2}\right)^{i}}{i!}\right) + \frac{3}{2} & x ...
2
votes
4answers
101 views

Direct Proof for sum of $n$ integers equation?

I am trying to prove by direct proof that $$3+5+7+\ldots+(2n+1)=n(n+2)$$ for all natural numbers $n$. I figured out how to do it by induction, but I know it can be done directly and I can't ...
-5
votes
1answer
38 views

Does $ \sum_{i=1}^\infty a_i \log a_i $ converge? [closed]

Let $a_1,\dots,a_n,\dots$ be a sequence of positive numbers such that $\sum a_i =1$. Does $$\sum_{i=1}^\infty a_i \log_2 a_i$$ converge?
0
votes
0answers
15 views

expression for a rational number

Let $x=\sum_{k=1}^n-b_kg^{-k}$ where $g\in \Bbb N$ and $b_k\in\{0,\cdots,g-1\}$. I want to write $x$ in the form $x=m+\sum_{k=1}^nc_kg^{-k}$ with $c_k\in\{0,\cdots, g-1\}$ and $m\in \Bbb Z$. Doing ...
0
votes
0answers
11 views

Can you get the average order of $ \left( 1+|\mu(n)| \right)^{M(n)} $, where $\mu(n)$ and $M(n)$ are the Möbius and Mertens functions, respectively

When yesterday I was interested in do a little study about the arithmetic function $$f(n)=\left( 1+|\mu(n)| \right)^{M(n)},$$ defined for integers $n\geq 1$, which $\mu(n)$ is the Möbius function and ...
1
vote
2answers
31 views

Find general formula for $\sum _{i=1}^{n} \frac {(-1)^i i}{(2i-1)(2i+1)}$

I was able to find formulas for simpler expressions but I can't find the general formula for this one: $\sum _{i=1}^{n} \frac {(-1)^i i}{(2i-1)(2i+1)}$ I don't see any particular trend that would ...
0
votes
2answers
23 views

General formula for a summation

I can't find the general formula for the following sum. $q \in \Bbb R, q \ne 1$ $\sum _{i=0}^{n} q^{2i}$ Any hints?
1
vote
1answer
34 views

How to expand the summation term with power?

How to expand the following: $$ \left( \sum^{M}_{m=0} \frac{x^{m}}{m!} \right)^{K} $$ where $M$ and $K$ are positive integers.
3
votes
0answers
145 views

Is there a way to write this recurrence relation in a simpler way (or a way that's faster to program)?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
0
votes
1answer
73 views

$\sum_{n=1}^{\infty} ne^{-2n}$ estimate to 4 decimal places

I am supposed to estimate the sum correct to 4 decimal places and assume it converges. I know that I am supposed to plug in numbers for $n$ (Instructor says that solving for $n$ is impossible) however ...