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

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0answers
25 views

Sum of powered numbers

Here is what I have achieved so far: S(n,p)= 1^p+2^p...+n^p By arranging it this way: ...
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3answers
83 views

evaluate the sum $\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$

I'm trying to evaluate this sum $$\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$$ I have no idea how to deal with it. With one sum I can, with partial-fraction decomposition, ...
2
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2answers
43 views

Nested sum $\sum_{i<j< \cdots < k} ij \cdots k$

I am wondering if there is any known closed form for the following nested sum? : $$ \sum_{i<j<\cdots <k} ij\cdots k $$ where each $i,j,\cdots,k =1, \cdots, n$ I tried the first one: $$ ...
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2answers
46 views

Alternating Sum of Cubes [on hold]

How is it possible to evaluate: $$\sum_{k=1}^n{((-1)^{n-k}\cdot k^3)}=n^3 - (n-1)^3 + (n - 2)^3 - \cdots \pm 1^3$$ The fact that there is the $\pm$ at the end makes it difficult.
2
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1answer
37 views

How do we derive the sum of $3^n$ and $2^n$

I know that $\quad\sum2^n = 2 (2^n-1)$ How can we derive this summation? And also how can we deduce the summation of $3^n$ from this ? I did observe this pattern : $$ \begin{align} n &= 1 ;\ ...
4
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2answers
33 views

Multiplication Principle and Inclusion-Exclusion: $2^n = \sum_{i = 0}^n (-1)^i \binom{n}{i} \binom{2n - 2i}{n - 2i}$

I began to compose an unnecessarily complicated answer to this question: If we had 25 people all who have 2 different balls, how would you work out how many combinations there would be if we want ...
1
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1answer
35 views

Understanding summations with Poisson

I'm currently doing a problem on Poisson processes and I've encountered the situation where I'm not sure why this summation is expanded as follows: And similarly I have tried expanding out the ...
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1answer
41 views

Exact value of a sum involving harmonic numbers

Could somebody tell me the exact value of this series? $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(5)}}{k} $$ where $$ H_k^{(n)}=\sum_{i=1}^{k}\frac{1}{i^n} $$ Thanks!
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1answer
22 views

Factoring constant in summation

Trying to show that adding a constant c to $\sum_{k=0}^\infty a^kx_k$, where a is a constant will just add some constant k to the summation eg. $$\sum_{n=0}^\infty a^n(x_n+c)$$ $$=k + ...
2
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4answers
61 views

How can I simplify $1\times 2 + 2 \times 3 + .. + (n-1) \times n$ progression? [on hold]

I have a progression that goes like this: $$1\times 2 + 2 \times 3 + .. + (n-1) \times n$$ Is there a way I can simplify it?
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1answer
26 views

Expanding a term with a sum

We have the following quantity: $$E\left[\left(\sum^n_{j=1} (X(t_j) - X(t_{j-1}))^2-t\right)^2\right]$$ My textbook says this can be expanded in the following way (colors are my touch) ...
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2answers
70 views

Closed form of sum with binomial

I want to find closed form of the following expression : $$\sum\limits_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{2k+1}$$ I have no idea how to do it.
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2answers
30 views

Approximate summation of flooring function

I have this summation: $\sum_{k=0}^x \lfloor{\frac{k}{c}}\rfloor$ Do you have any ideas on any general expressions that can approximate this? P.S I know I can approximate it with a Fourier ...
1
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1answer
36 views

Sum of the series with Stirling numbers of the first kind.

Yesterday I worked on one problem in discrete math and in the process of decision I came across this series. Try to do it with generating functions, but there is no success for me. So, what do you ...
0
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1answer
27 views

Summation operation for precalculus

Studying Spivak's Calculus I came across a relation I find hard to grasp. In particular, I want to understand it without using proofs by induction. So please prove or explain the following ...
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1answer
26 views

Sums involving exponential functions

I am trying to find the closed form of the following related sums: $$(i)\quad\quad S_1(n)= \sum_{m=-\infty}^{m=\infty} |n-m| e^{-p(|n-m|+|m|)} $$ $$ (ii)\quad\quad S_2(n)= ...
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5answers
122 views

How to prove $\sum\limits_{i=0}^n (-1)^i \binom{n}{i} \binom{n-i}{k}=0$

I would like to prove that: \begin{equation*} \sum\limits_{i=0}^n (-1)^i \binom{n}{i} \binom{n-i}{k}=0;~k\geq0 ; n\geq1. \end{equation*} Can any one help me how to do that? Thanks
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1answer
33 views

Sigma sign problem from Spivak's calculus text ch 2-2

I need to find a formula for $$ \sum_{i=1}^n (2i-1)^2 = 1^2 + 3^2 + \cdots + (2n-1)^2 $$ This problem is contained in Spivak's calculus ch2-2. I know that: $$ \sum_{i=1}^n i^2 = ...
1
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1answer
28 views

Looking for tip/procedure of series solutions to ODE

I have been having a few questions about series solutions to ODE and I found an example that can illustrate my question. It is just a simple example, say we consider the ODE $$ y''-xy'-y=0$$ around ...
3
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1answer
63 views

Proving that a trigonometric sum is in $L^2$

How can I use Parseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? Thank you!
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0answers
24 views

Estimate from above $\sum_{m=1}^{n-1}\frac{1}{n^\alpha-m^\alpha}$

Find an upper bound for $$\sum_{m=1}^{n-1}\frac{1}{n^\alpha-m^\alpha}$$ with $\alpha>1$. I do not know where to start but, for example, if $\alpha=1$ the previous sum is linked to Harmonic numbers ...
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1answer
24 views

Question about index of summations

I have a question in regard to changing the index of summation. For example, I am confused on why for some problems in my ODE class , such as ' $y''-y'=0$ for example, we suppose the summation of ...
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1answer
38 views

Find the value of $\sum^{n-1}_{m=1}\left(\frac{1}{n-m}+\frac{1}{n+m}\right)$.

Find the value of $$\sum^{n-1}_{m=1}\left(\frac{1}{n-m}+\frac{1}{n+m}\right)$$ I used WolframAlpha obtaining $$\psi^{(0)}(2n)-\frac{1}{n}+\gamma$$ where $\gamma$ is the Euler-Mascheroni constant and ...
7
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0answers
62 views

In how many ways can the integers from $1$ to $n$ be divided into two groups with the same sum?

In how many ways can the integers $1,2,\ldots,n$ be divided into two groups with the same sum? I have tried calculating some of these values for small $n$, but cannot seem to find a pattern. Any ...
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1answer
29 views

On finite sums and products

I'd like to get a good book on finite summations and products before I study infinite series more in depth next year. The book should cover geometric/ harmonic sums and prove different formulas for ...
3
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1answer
57 views

Sophomore's dream changing “x”

"Sophomore's Dream" says $\sum_{n=1}^{\infty}n^{-n}=\int_0^1x^{-x}$ Can you replace the $x$ and $n$ with $2x$ or $x^3$ (and $2n$ or $n^3$) or something? I would guess not, because replacing $x$ with ...
6
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2answers
137 views

Proof of an identity of $n!$

I came up (numerically) with an identity concerning n! and I was wondering about a proof of it. Here it is: \begin{align} \ n! &= \sum_{r=0}^{n} { \binom{n}{r} (-1)^r(k-r)^n } \quad \forall n ...
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1answer
26 views

Differentiating expression involving summation

My problem seemed very simple at glance but I keep missing one term from the answer. Any suggestions? This is the problem: We have $$x_i^* + \xi_i + \frac{\alpha_i}{p_i} \left[ y - \sum_{j=1}^n ...
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1answer
28 views

Differentiating a sum involving logs

I was doing the problem provided in the picture but I do not understand how do they obtain the answer. I am not sure how to differentiate the sum. I end up getting: alpha - 1 - 1/K. I believe I need ...
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1answer
24 views

How to express outer sum in a matrix form?

So I have the following equation for a matrix $\mathbf{B}$ given $\mathbf{A}$: $$ b_{ij} = \sum_k \sum_l a_{ki} a_{jl} $$ The question is if there is anyway that I can write that one compactly in ...
6
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2answers
401 views

Integrating over the naturals. What does it mean?

Let $F$ be the power set of $\Bbb{N}$ and consider the measurable space $(\Bbb{N}, F)$. Then what does it mean to take the integral with respect to the measure $\mu(A) = \sum_{a \in A} \frac{1}{a}$. ...
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1answer
19 views

“Sum of power” for prime numbers

I use Euler–Maclaurin formula, Faulhaber's formula and Bernoulli polynomials for "sum of powers" for this type $\sum_{t=1}^nt^k$. but I don't know to find compact form when sum is taken from first ...
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1answer
26 views

What is the difference between finding the sum of a series and its closed-form solution?

In complexity theory, it is sometimes necessary to find the "closed-form solution" of a summation. This was put in our exam guide as "solving arithmetic and geometric series", which I initially ...
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0answers
23 views

Find $Z$ transform of given signal

Given the discrete signal $h(n)=r^n\frac{\sin{[(n+1)\theta]}}{\sin{\theta}}$ if $n \geq 0$ and $h(n)=0$ otherwise, find the $Z$ transform of $h(n)$. What I did: We know that ...
2
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1answer
57 views

Looking for formula of $\sum_{k=1}^m (-1)^k \dfrac {x^2(x^2-1)…(x^2-k+1)}{(x+1)(x+2)…(x+k)}$

Let \begin{equation*} u_k:=(-1)^k \dfrac {x^2(x^2-1)...(x^2-k+1)}{(x+1)(x+2)...(x+k)}. \end{equation*} Can we find the sum of first $m$ of $u_k$ 's? That is, is there any formula for $\sum _{k=1}^m ...
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0answers
15 views

Estimation for a logarithmic function in $(0,\,1)$. A series should be used?

Let $f(t)\geq C_1t^{-\alpha}$ for all $t\in(0,\,\infty)$ and for some $C_1>0,\,\alpha>0$. and let $g(t)\geq C_2\left(\ln(t^{-1})\right)^\beta$ for all $t\in(0,\,1)$ and for some ...
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4answers
73 views

Prove that $\sum\limits_{r \mathop= 0}^n \frac {(-1)^r} {r!(n-r)!} = 0$

I wish to prove that $\displaystyle \sum_{r \mathop= 0}^n \frac {(-1)^r} {r!(n-r)!} = 0$ It is plain when $n$ is an odd integer. How might one go about proving it generally?
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2answers
61 views

Help with sequence $1\cdot n + 2(n-1) + \ldots + (n-1)2 + n\cdot 1$

Can anyone please provide a simplified formula for the sum of the sequence \begin{equation*} s(n) = 1\cdot n + 2(n-1) + \ldots + (n-1)2 + n\cdot1 \end{equation*} where $n$ is an integer greater than ...
2
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1answer
77 views

Other variation of Nicomachus's Theorem?

We all know that $ 1^3+2^3+3^3 + \ldots + n^3 = (1+2+3+\ldots + n)^2 $. Denote $\displaystyle S_m = \sum_{j=1}^n j^m $, then we can set $ S_3 = S_1 ^2 $ for all positive integers $ n $. Question: Is ...
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0answers
14 views

Abel's summation formula for functions depending on limit of sum

Abel's summation formula states that for two functions $f$ and $g$, with $f$ differentiable, we have $$\sum_{k=1}^n f(k)g(k)=G(n)f(n)-\int_1^n G(x)f'(x)\; dx \tag{$*$}$$ where $G(n)=\sum_{k=1}^n g(k)$ ...
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1answer
21 views

Proof of Lagrange Identity

I need to prove Lagrange Identity for complex case, i.e. $$ \left( \sum_{i=1}^n|a_i|^2 \right)\left( \sum_{i=1}^n |b_i|^2 \right)-\left| \sum_{i=1}^na_ib_i \right|^2=\sum_{1\leq i<j\leq ...
4
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2answers
48 views

Prove Lagrange's Identity without induction

Prove Lagrange's Identity without induction. $$ \sum_{1\leq j <k\leq n}(a_jb_k-a_kb_j)^2=\left( \sum_{k=1}^na_k^2 \right)\left( \sum_{k=1}^n b_k^2 \right)-\left( \sum_{k=1}^na_kb_k \right)^2 $$ I ...
4
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0answers
34 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
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4answers
37 views

How do I read this triple summation? $\sum_{1\leq i < j < k \leq 4}a_{ijk}$

How do I read this triple summation? $$\sum_{1\leq i < j < k \leq 4}a_{ijk}$$ The exercise asks me to express it as three sumations and to expand them in the following way: 1) Summing first ...
2
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1answer
56 views

Average of elements in a subset of $\{1,2,3,..,n\}$ is greater than $\frac{n+1}{2}$ [closed]

Consider two integers $n \ge m \ge 4$ and $A=\{a_1,a_2,...,a_m\}$ a subset of the set $\{1,2,3,...,n\}$ with the property that $$\forall a,b \in A \text{ with } a \neq b, \text{ if } a+b \le n, \text{ ...
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2answers
32 views

Closed form formula for discrete sums [closed]

Is there a general way to obtain a closed form formula for any discrete sum of the form: $\sum_{a}^{b}f(n)$ with certain restrictions on the form of $f(n)$, much like how we can find closed form ...
0
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0answers
26 views

Sum of products of K numbers taken from N numbers in closed form

Let's say i have 5 numbers, $A,B,C,D,E$. I want to know the sum of all the possible products of some or all of these numbers each taken at most once. Instead of a lot of multiplications and additions ...
3
votes
0answers
41 views

Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
0
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1answer
26 views

Is it possible to exhibit a collection of sets

Let a subset $D$ of the natural numbers be called convergent or divergent when the associated series $\sum_{d \in D} \frac{1}{d}$ converges or diverges. Define a topology on $\Bbb{N}$ by defining the ...
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0answers
53 views

Do sum indices need to be integers?

I have this exercise: Compare if these two sums are equal: $\sum_{k=0}^5a_k$ and $\sum_{k^2=0}^5a_{k^2}$ I know the first one is $a_0+a_1+\cdots+a_5$, but I'm wondering if the second one is the ...