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

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1answer
27 views

Convergence of the series - best criertion

What will be the best criterion to use to investigate convergence of the series (i do not need step by step explaination) $$\sum_{n=1}^\infty \frac{e^{\frac{1}{n}} }{n^{e}}$$
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0answers
59 views

double sum with a binary variable of three elements

I have a binary variable $\ v(s,c,h)\ $which takes value 1 if subject $\ s\ $is taught in classroom $\ c\ $in time slot $\ h\ $ and 0 otherwise. I have a question about a type of constraint that I ...
2
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0answers
35 views

Einstein summation convention

I am not sure how to expand the following expression with regard to the Einstein summation convention. More specifically, I have: \begin{equation} a_{ij} = b_{i, j} + b_{j, i} + c_{ij, kk} \\ c_{ij} = ...
1
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1answer
46 views

Partitioning a set to get a sum

I have a set of numbers: 2,2,4,4,4,4,4,4,6,6,6,6,6,6 I want to enumerate the possible ways to partition this set into 4 groups, each of which sum to 16. How can I approach this short of brute force? ...
3
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0answers
44 views

Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
1
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1answer
56 views

Derivative of a summation function in order to minimize the function

I'm asked to minimize this function $$f\left(x\right)= \sum_{k=1}^K \left(g\left(w\left(k\right)+\alpha\right)-t\left(k\right)\right)^2$$ with respect only to $\alpha$. Function ...
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0answers
22 views

Double summation probability [duplicate]

We define $X_i = \mathbb{1}_{\{Z_i=0,Z_{i-1}=1\}} = \mathbb{1}_{\{Z_i=0\}}\mathbb{1}_{\{Z_{i-1}=1\}}$ ($\mathbb{1}_E$ is the indicator of $E$), so that $U_n=\sum_{i=2}^n X_i$, and know that ...
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2answers
43 views

How to solve the following recurrence

I know others have already posted about this recurrence $T(n) = 2T(n/2) + n\lg n$ on the following these two posts: post1 and post2 However, the style in which they have solved them, is not one with ...
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2answers
27 views

Discrete Math-Computing Summations

So I'm asked to compute a summation with an upper limit $k = 20$ and lower limit $k=1$, where: $B_k= 0$ when $k=1$, and $B_k = \dfrac{1}{(k^2-1)}$ , for $k>1$. I was wondering if there is a ...
0
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1answer
54 views

Prove this sum is always zero

I encountered this problem in the context of singular homology, trying to prove that the boundary map is always zero. How do you show that the following sum, for all $n\in\mathbb{N}$, is always zero? ...
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2answers
28 views

Trying to find a formula for the following algorithm

I am trying to make a formula for the following algorithm as a function of n, building up my answer using summations. The algorithm is: ...
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4answers
20 views

Selection Sort Summation Simplification

I am trying to simplify the summation for selection sort. Starting out with: $$\sum_{i=0}^{n-1}\sum_{j=i+1}^{n-1}1$$ I am able to get: $$\sum_{i=0}^{n-1}n-i-1$$ However, I don't understand how to ...
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1answer
38 views

Summation of 1/n^2 using Fourier series on different intervals

I have been going through my notes on complex Fourier series and came across the following anomaly which I hope someone can help me with. I calculated the complex Fourier series for the function ...
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2answers
57 views

Evaluate $\sum_{n=1}^\infty \frac{n}{2^n}$.

Evaluate $$\sum_{n=1}^\infty \frac{n}{2^n}$$ My Work: $$\sum_{n=1}^\infty \frac{n}{2^n} = \sum_{n=1}^\infty n \left(\frac{1}{2}\right)^n$$ If we denote $f(x) = \sum_{n=1}^\infty nx^n$ then we ...
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0answers
18 views

How to sum random variables

Let $Z_t = \psi_t |\lambda Z_{(t-1)} + (1-\lambda)\epsilon_t |$ be a random variable where $\epsilon~N(0,1)$ is a Gaussian distributed number, $Z_0 = z_0$ and $\psi \in [-1,1]$ a random variable, ...
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0answers
40 views

How would I put these recurrence relation terms into a summation?

I was given these terms as part of a recurrence relation and I need to put it into a summation in order to solve it. $T(n)=2^{k}T\left(\dfrac{n}{2^{k}}\right) + 2^{k-1}T\left(\dfrac{n}{2^{k-1}}\right) ...
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1answer
27 views

Is there a formula for the summation of this form?

I am doing recurrence relations and I have done some work to get the summation $$\sum\limits_{i=0}^{k-1}16^{i}\left(\dfrac{n}{4^i}\right)^2.$$ I know that there is a formula if the summation was just ...
0
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1answer
52 views

Manipulation of summations

this question branches off another question that can be seen here Now we begin be taking a look at the following expressions: $$ \sum_{k=1}^{n-l} \sum_{j-0}^m \frac{\ln(g)^{m-j}}{g^k} ...
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0answers
17 views

Continued Fraction summation representation

I have a rational fraction of the form: $$s=\frac{p_0+p_1x+p_2x^2+\cdots+p_Mx^M}{1+q_1x+q_2x^2+\cdots+q_Mx^M} $$ The paper I am reading converts this to the form: $$s = ...
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1answer
28 views

Methods for Improving Convergence of a sequence of Partial Sums

I have the following sum: $$\zeta(3)+\frac1{4}=\sum_{k=0}^{\infty}\frac{2k^2+7k+7}{(k+1)^3(k+2)(k+3)}$$ Are there any methods that I can use to speed up the convergence of the sequence generated by ...
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2answers
48 views

finding the sum function of $\sum_ {n=1}^{\infty} \frac{n-2}{(n-1)!} z^{n+1}$

finding the sum function of $\sum_ {n=1}^{\infty} \frac{n-2}{(n-1)!} z^{n+1}$ So far i've substituted n-1 for m which gives me the following form: $\sum_ {m=0}^{\infty} \frac{m-1}{(m)!} z^{m+2}$. ...
1
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1answer
37 views

Multi-index power series

What is closed-form expression for the summation $$ S(n,m)=\sum_{|\alpha|=m} p^{\alpha} = \sum_{\alpha_1 + \cdots + \alpha_n = m} \prod_{i=1}^n p_i^{\alpha_i} $$ as a function of $n$ and $m$? Here ...
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2answers
21 views

Need help understanding partial solution for sum $\sum_{1\leq i \leq j \leq n}(j-i)$

Let's consider the following sum: $$\sum_{1\leq i \leq j \leq n}(j-i)$$ Here are some progressions from my Discrete Math lecture: $$\sum_{1\leq i \leq j \leq ...
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2answers
79 views

Integration of 1/x as a limit of a sum

This is from R.Courant book Example "Introduction to Calculus and Analysis vol.1 " To integrate $x^\alpha$ when $\alpha\neq1$ we subdivide the interval [a,b] by the point of geometric progression: ...
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1answer
35 views

Itemwise Proportion

I have to calculate efficiency of some machines in our factory. The calculation process is per roll production base, that means when a roll comes out, we calculate efficiency of that roll. These ...
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5answers
81 views

Is $\sum_ix_iy_i=\sum_ix_i\sum_iy_i$?

I ask this because the equation for the center of mass of a system (made up of a number of small masses attached to each other) is given by: $$\bar x=\frac{\sum_im_ix_i}{\sum_im_i}$$ If the ...
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0answers
12 views

Help unpacking this system of differential equations

The geodesic equation is given as $~~\ddot{\alpha} + (\dot{\alpha} \cdot N \dot{\circ} \alpha)(N \circ \alpha) = 0$, where $N(p), \alpha(t) \in \mathbb{R^{n+1}}$ and $p \in \mathbb{R^{n+1}}$, $t \in ...
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1answer
30 views

What is the correct name for a “summable” number?

My math/CS teacher mentioned a function to me a few days ago (I don't remember the context), but didn't know the real name for it, so he just called it a summable function. We didn't really go into ...
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1answer
59 views

How can one differentiate with respect to variable upperbounds in summations?

I have been looking at derivatives of the form: $$\frac{d}{dx}\sum_{i=1}^{x}f(i).$$ There is a simplification in the definition of such a derivative: $$\frac{d}{dx}\sum_{i=1}^{x}f(i)=\lim_{h\to ...
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3answers
95 views

Sum of powers: $1^m+2^m+3^m+…+n^m$=? [closed]

For any positive integer $n$ and $m,$ I was wondering if there is any way to get a closed formula for $$S(n,m)=1^m+2^m+3^m+\cdots+n^m$$ something like $$S(n,1)=1+2+3+\cdots+n=\frac{n(n+1)}{2}.$$
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2answers
43 views

How to represent the following in sum notation,also notify if this is any special series?

$\color{blue}{1.}~~ -x+\frac{x^3}{3}-\frac{x^5}{5}+\frac{x^7}{7}\cdots \infty$ and $\color{blue}{2.}~~ \frac{x^2}{2}-\frac{x^4}{4}+\frac{x^6}{6}-\frac{x^8}{8}\cdots \infty$ i know $1,3,5,7\cdots$ ...
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1answer
45 views

prove that $\lim \limits_{n \rightarrow \infty} n \sum \limits_{j=1}^{n} \frac{cos(\frac{n}{j}) f(\frac{n}{j})}{j^2}$ exists and final.

$f$ is monotonically decreasing function such that $\lim \limits_{x \rightarrow\infty} f(x) =0$, prove that the following limit exists and final . $$\lim \limits_{n \rightarrow \infty} n \sum ...
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0answers
72 views

Calculation of an expression ($\max_{U}\min_i \sum_j |U_{ij}|^2 |e_i^j|^2$)

There is an orthonormal basis $\{e_i\}(i=1,\ldots,n)$ in $\mathbb{C}^n$, each of them is represented in form of column vectors $$\begin{pmatrix} e_i^1\\ \vdots\\e_i^n\end{pmatrix}.$$ My purpose is to ...
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2answers
54 views

Prove, that the sum difference of all consecutive prime numbers from $p_1$ to $p_n$ is $p_n-p_1$

Example: $\mid (2-3)+(3-5)+(5-7)+(7-11)\mid =11-2=9$ I tried a couple of basic tricks to reach some proof but I failed.
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0answers
25 views

Gradient of a summation

How to calculate the gradient of the following summation in terms of $x_i$ ? $$\sum_{i=0}^n(x_i-a)^2$$ is the following answer true? $$2 \sum_{i=0}^n(x_i-a)$$ Thank you
4
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1answer
82 views

Interchange summation and differentiation for ONB

Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$ Now assume we have that $$\frac{d}{dx}e_n = \lambda_n e_n.$$ Assume $f \in H^1[0,1],$ so i.e. $||f'||_{L^2} < \infty$ ...
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0answers
48 views

Summation by parts with only one variable

I have a question about summation by parts: I am supposed to show that $\sum_{j=1}^{n}(v_j^m)(v_{j-1}^m-2v_j^m+v_{j+1}^m)=-\sum_{j=1}^{n}(v_{j+1}^m-v_j^m)^2$ where m is dependent on time. I am given ...
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0answers
8 views

Confused about a step with the summation operator.

Image 1 contains the step which I am confused about. What happens to the middle term (-XiY(bar) -YiX(bar)). Picture 2 contains the question for context. I understand how to do the question once I ...
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2answers
59 views

Prove, If the sum of the first $n$ prime is also a prime then it is also a hypotenuse of a primitive Pythagorean triples

I checked this for all the primitive Pythagorean triples $<300$. Some examples would be: a. $2+3=5$, b. ...
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2answers
32 views

Find the Harmonic Mean

The question: Peter drives to work, a distance of 50 miles, at a speed of 75 mph and returns home at a speed of 80 mph. What is his average speed for the round trip? The formula to use here is ...
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0answers
18 views

If $s_{k,m}(n) =\sum_{i=n+1}^{kn+m} \frac1{i} $ show that for $k \ge 2m+1$, $s_{k,m}(n+1)>s_{k,m}(n)$ and $s_{k,m}(n+1)-s_{k,m}(n) <\frac1{n(n+1)} $

Let $s_{k,m}(n) =\sum\limits_{i=n+1}^{kn+m} \frac1{i} $. Show that, for $k \ge 2m+1$, $s_{k,m}(n+1)>s_{k,m}(n)$ and $s_{k,m}(n+1)-s_{k,m}(n) <\frac1{n(n+1)} $ so that $s_{k,m}(n) < ...
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1answer
28 views

Application: Sum of Digits

if a five digit number N is such that sum of its digit is 29, can N be square of an integer? Suppose N be abcde, where a+b+c+d+e = 29. Can square of any number less than abcde is equal to abcde ...
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2answers
45 views

How to deduce sum's result?

I was solving some tricky-task with algorithms and I obtained following reccurence: $$a_{k+1} = 4a_{k} + 16^{k}, a_{1} = 1$$ It's obvious that with given start condition: $$a_{k+1} = 4a_{k} + 16^{k} ...
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4answers
36 views

How to computer the summation of a binomial coefficient/ show the following is true

$\sum\limits_{k=0}^n \left(2k+1\right) \dbinom{n}{k} = 2^n\left(n+1\right)$. I know that you have to use the binomial coefficient, but I'm not sure how to manipulate the original summation to make ...
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4answers
166 views

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$ $$\begin{align}\frac{1}{n^4+n^2+1}& =\frac{1}{n^4+2n^2+1-n^2}\\ &=\frac{1}{(n^2+1)^2-n^2}\\ &=\frac{1}{(n^2+n+1)(n^2-n+1)}\\ ...
1
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2answers
44 views

Probability that colored balls are separated

Say we throw $b$ blue balls and $r$ red balls uniformly into $n$ boxes. The probability that no box contains a red as well as a blue ball is then, by the inclusion exclusion principle: $$p = ...
3
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1answer
117 views

Inclusion relation between two summability methods

Let $0\leq x<1$ and $s_n$ be a sequence of partial sums of the series $\sum_{n=0}^{\infty}a_n$. It is called that the series $\sum_{n=0}^{\infty}a_n$ is $(A)$ or Abel summable to $s$ if ...
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2answers
69 views

Closed Form for Finite Sum: Product of two Similar Functions

I need to find a closed form expression in terms of $c$, $n$, $x$ and $y$ for $$ \sum_{j=0}^{n}\rho^{c-j}\frac{x^j}{j!}\frac{y^{c-2j}}{\left(c-2j\right)!} $$ where $c$ and $\rho$ are just constants. ...
0
votes
2answers
41 views

Reciprocal squares sum inequality [duplicate]

What is the easiest (preferably inductional) way without approximation of the sum_ to prove the following inequality: $\frac{1}{1^2}+\frac{1}{2^2} + \ldots +\frac{1}{n^2} \le 2 - \frac{1}{n}$
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0answers
32 views

Sum of exponents equals delta function

For a homework exercise i need to prove that $$\sum_{n\in\mathbb{Z}} e^{i(k-\xi)n}|u(\xi)> =\delta(k-\xi)|u(\xi)>.$$ Where $\delta(x)=1$ if $x=0$ and zero elsewhere and $|u(\xi)>$ is just a ...