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Questions tagged [summation]

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

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Minimal conditions to show $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below $$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$ $$G = ...
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130 views

Asymptotic solutions of a sparsely perturbed recurrence relation

Recurrence relation I am trying to find approximate solutions $T(n)$ of the recurrence relation $$ p\ T(n-1) - \left[p+q+\overline{S} + \varepsilon \tilde{S}(n)\right]T(n) + q\ T(n+1) = 0,\\ \text{...
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44 views

Looking for a clever simplification of $\sum_{j=2}^{T_L}(j-1-O_L)(2j-n-1)+\sum_{i=T^U}^{n-1}(O^U-n+i)(2i-n-1)$

I have the expression (which I got by adding up all the distances between $n$ natural numbers bounded by $a_n$ and $a_1$): $$\sum_{j=2}^{T_L}(j-1-O_L)(2j-n-1)+\sum_{i=T^U}^{n-1}(O^U-n+i)(2i-n-1)$$ ...
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175 views

Richert’s theorem breaks down for $ n = 11 $

In 1949 H.-E.Richert proved (1) that every positive integer typeset structure is a sum of distinct primes. For more information please look at (2), and (3). However, if you consider $ n = 11 $, you ...
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110 views

Finding a closed form for this summation

I have been trying to derive a few identities using some bell polynomials and a technique i have come up with and i came across this summation: $$ \rho(n,k) = \sum_{j=0}^k {k \choose j} {\frac{-j}{2} ...
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93 views

Value of double sum of powers of fractions between 0 and 1

Is there any way to find closed form for the sum (where k is positive integer) $$S = \sum_{i = 1}^{n}\sum_{j = 0}^{i} \left( \frac{j}{i} \right) ^ k$$ Using Faulhaber's formula I got $$S = \frac{1}{...
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83 views

Bizarre differential identity.

Let $d\ge 1$ be an integer. Let $m$ and $n$ be integers subject to $m \ge n+d-1$. The question is to prove the following identity. \begin{equation} \sum\limits_{j=-1}^{d-1} \sum\limits_{\underset{d_1,...
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221 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 ...
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107 views

Trouble writing Double Summation

I have the following: $$\left(\frac{C_0}{0!}a_m+\frac{C_1}{1!}a_{m-1}+...+\frac{C_m}{m!}a_0\right)x^m+\left(\frac{C_1}{1!}a_m+\frac{C_2}{2!}a_{m-1}+...+\frac{C_{m+1}}{(m+1)!}a_0\right)x^{m+1}+...+\...
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80 views

How to prove that : $\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
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51 views

Is it possible to eliminate the inner sum to evaluate numerically?

Any hints on how to simplify the following double sum to be able to find the sum at least numerically? $$\sum_{n=2}^{\infty}\frac1{n(n^2-1)} \sum_{k=1}^\infty \frac{(k-1/n)^{2n-2}}{(k+1/n)^{2n+2}}$$ ...
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287 views

Sum problem involving Factorials

got the following problem to prove for $n \in \mathbb{N}$ and $1 \leq i \leq n$: \begin{equation} \sum_{k=1}^{n-i} \frac{(2n-1-i-2k)! 2^{2k}}{(n-i-k)! (n-k)! 2}=\sum_{k=1}^{n-i} \frac{(2n-1-i-k)! 2^{...
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100 views

Asymptotics of integer compositions

A (weak) composition of a positive integer $n$ into $k$ parts is an ordered sequence of nonnegative integers $(a_1, a_2, \ldots, a_k)$ such that $ \sum_{i=1}^k a_i = n $. I am interested in the case ...
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180 views

Finding the length of an elliptical spiral

Okay, i had a very strange thought, it was "Is it possible to find the length of an elliptical spiral whose major and minor axes were decreasing?" Like for example lets say that $$ \frac{a}{b} = n ...
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87 views

Can this summation be expressed differently?

Lets say I have a sum that states the following $$ \sum_{j=0}^{k-c} {k-c \choose j}\ln(a)^{k-c-j} \frac{d^j}{dx^j}[(x)_c] $$ where $(x)_c$ is the falling factorial such that $$ (x)_c = x(x-1)(x-2)\...
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97 views

How to find $\sum_{k=1}^n k^k$?

Actually question which I found: Find the sum of the series $1^1+2^2+3^3+ \cdots +n^n $ This question has been bothering me since a long time. Any help would be appreciated!
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36 views

Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum $$\...
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69 views

Interchanging index of summation in $d$ dimensions

Let $\alpha = (\alpha_{1}, \ldots, \alpha_{d}) \in \mathbb{Z}_{\geq 0}^{d}$ and let $|\alpha| = \alpha_{1} + \cdots + \alpha_{d}$. I have the following question about interchanging summations: Is $$\...
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203 views

Evaluation of the double sum

Is there a way to get a closed expression for the double sum: $$\sum\limits_{n = 1}^\infty \sum\limits_{m = 1}^\infty \left( \frac{m}{(n^2 + m^2 - E_1)(n^2 + m^2 - E_2)} \right)^2$$ , where $E_1$ and $...
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158 views

Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq i}^n\frac{\sin(\theta_j-\theta_i)}{\left(1-\cos(\theta_j-\theta_i)\right)...
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122 views

Interchange of limiting operations (question from an engineer)

I need to clarify when are the below operations valid. If possible, please link me to the related theorems, where I can find details. 1- Given a double integral \begin{equation} \int_{X}\int_Y f(x,y)...
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234 views

Finding sum for function with floor-function

I am trying to find a formula to calculate the following sum: $$\sum_{x=0}^n (2ax - {1 \over 2}a^2 - {1 \over 2} a) $$ where $$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$ and $$\phi = {1 + ...
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235 views

Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
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143 views

How to prove this combinatorial identity

I am wondering how to prove the following identity: $$\sum_{k=0}^r {r-k \choose m} {s \choose k-t} (-1)^{k-t} = {r-t-s \choose r-t-m}$$ It seems that I can negating the upper index of ${s \choose k-t} ...
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104 views

Find Gcd summation fast?

Find the value of the summation: $$ val=\left( \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^c....\sum_{x=1}^p GCD(i,j,k,..x) \right)$$ Contraints $2\leq$number of summation terms$\leq 500$, $1\leq a,b,c.......
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189 views

Product of Summations for All Subsets

We have a set $X$ of $n$ integers $\{$$x_1$, $x_2$, .. , $x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, ...
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75 views

Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online ...
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834 views

Expectation identity interchange of infinite sums

I was deriving the identity $$EX = \sum_{n=1}^{\infty} p(X\geq n).$$ I had to use an interchange of limit sums from $\sum_{n=1}^{\infty} \sum_{i=n}^{\infty} p(X=i)$ to $\sum_{i=1}^{\infty} \sum_{n=1}^...
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741 views

Proving that the finite sum of the each reciprocal of any sequence of integers with common difference is not an integer.

Question : Could you show me how to prove that $\sum_{j=1}^{n}\frac{1}{a+jd}$ is not an integer for any integers $a\gt1, d\gt0$. A week ago, I found the following question in a book: Prove that $S_n=...
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849 views

Why does the following equation hold?

$\sum_1^\infty\frac{(k\theta e^{-\theta})^k}{k!}=\frac{\theta}{1-\theta}$, where $0<\theta<1$. It can be verified via simulation, but I haven't proved it. Are there any previous results on ...
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47 views

For what $f$ is it true that $\lim_{n\to\infty}\sum_{k=0}^nf(n,k)=\sum_{k=0}^{\infty}\lim_{n\to\infty}f(n,k)$

Let $f:\mathbb{N_0}^2\to\mathbb{R}$. For what $f$ is it true that $$\lim_{n\to\infty}\sum_{k=0}^nf(n,k)=\sum_{k=0}^{\infty}\lim_{n\to\infty}f(n,k):=\lim_{m\to\infty}\sum_{k=0}^m\lim_{n\to\infty}f(n,k)...
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231 views

Sum formula for the $\Omega$ constant

I was looking a bit around, and was interested in the konstant $\Omega$. It is defined as number satisfying the equation $$ x e^x = 1 $$ Now, Wikipedia, gives an reccurence relation for the constant ...
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174 views

Eliminating negative binomial coefficients

I have a sum $$\sum_{m=0}^{n}\sum_{j=0}^{k}(-1)^j{k \choose j}{n-mj \choose k}$$ that comes up when counting compositions. Now the trouble is, if I would interpret it literally and for $n<mj$ ...
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240 views

Proving identity involving sum

I'm stuck trying to prove the following identity: $$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 \...
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224 views

Bernoulli formula

The sum: $$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$ Can be calculated by this formula, called the "Bernoulli formula" in wikipedia $$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k n^{m+...
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194 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
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87 views

Is it possible to algebraically prove that the $n$th degree Taylor remainder of $f(x)$ is less than $K|\Delta x|^{n+1}$ for $K \in \mathbb{R^+}$?

I found a purely algebraic proof, given below, that for a mononomial $f(x) = x^n$ the magnitude of the error of its linear approximation $| f(x) - [f(a) + f'(a)(x-a)] |$ is less than $K(x-a)^2$ for $...
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184 views

Limit involving sums of the Von-Mangoldt function

Can someone show that the limit bellow approaches 1/2? Can you also prove that it does, with out using the prime number theorem? $$ \lim_{n\to\infty} \frac{\sum\limits_{k=1}^n \Lambda(k) \frac{k}{n}\...
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71 views

$\sum_{i=1}^x\sum_{j=1}^xf(i\cdot j)$ Double Summing a (Not Completely) Multiplicative Function

Let $f(n)$ be a multiplicative function that is not completely multiplicative, i.e $f(m)\cdot f(n)= f(m\cdot n)$ only if $gcd(m,n)=1$. Let $S(x)$ be the double sum over $f$, that is: $$S(x)=\sum_{i=1}...
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55 views

Is there a simpler way to sum the first $n$ terms of the sequence of numbers starting at 2 and squaring to get the next?

I've got this summation: $$ f(n)=\sum_{i=0}^{n-1}2^{2^i} $$ In effect, it's the sum of the sequence of numbers you get from starting at 2 and squaring the previous number in the sequence: $$ \begin{...
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120 views

Show that $\sum x^p$ over primes must have a non-trivial zero.

The sum $\sum x^n$ is unbounded in $|x| \le 1$. Similarly if $p$ is prime then trivially $\sum x^p$ is also unbounded in $|x| < 1$ because all primes $> 2$ are odd so the lower bound would ...
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32 views

Joint conditional probability of geometric and poisson distributions

I'm new to the concept of joint conditionals, and I want to make sure that a move I made is valid and logic. $$X\sim Geom(0.21) \rightarrow P(X=k)=(0.79)^{k-1}\times0.21$$ $$Y|X\sim Poisson(x+1)\...
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39 views

Sum of reciprocal sine function $\sum\limits_{k=1}^{n-1} \frac{1}{\sin(\frac{k\pi}{n})}=?$

The question comes to me when I find there are answers on summation of some forms of trigonometric functions, i.e. $$ \sum\limits_{k=1}^{n-1} \frac{1}{\sin^2(\frac{k\pi}{n})}\\ \sum\limits_{k=0}^{n-1}...
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0answers
27 views

Closed form and asymptotic solutionof $\sum_{a=1}^N \lfloor N/a \rfloor \lfloor (N \pm \lfloor N/a \rfloor)/a \rfloor$

I am solving polynomials with constraints on the coefficients and I get the following expression $$\sum_{a=1}^N \lfloor N/a \rfloor \lfloor (N \pm \lfloor N/a \rfloor)/a \rfloor$$. I am looking for ...
2
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0answers
73 views

A summation of the multiplication of reflected Mobius functions and their behavior for different values of $k$

$$ \Psi_k(N)=\sum_{n=1}^{N} \mu(n)\mu(k-n)$$ where $\mu(n)$ is the Mobius function. This function is interesting to me because for the case of $k=N$ it has the symmetric property of being odd with ...
2
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0answers
73 views

Summation of Integrals without using $\zeta(2)$

How do you evaluate the sum $$\sum_{k=1}^\infty\int_{\sqrt k}^{\sqrt{k+1}} \left(\frac{x^2}{k}-1\right)\, dx$$ without using $\zeta(2)$? I can see some relationship to $\dfrac1{\lfloor x^2\rfloor}$ ...
2
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0answers
97 views

A sort of Wolstenholme's $p^2$-congruence.

It is well-known (Wolstenholme Theorem) that for any prime $p$ such that $p>3$ the Harmonic number $H_{p-1}$ satisfies the congruence $$ H_{p-1}:= \sum_{i=1}^{p-1}\frac{1}{i}\equiv 0 \pmod {p^2}$$ ...
2
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0answers
26 views

Have I been rigorous in this simple argument with summation and expectation operators?

I am told that $A$ and $B$ are random variables, and that $\mathbf{E}(A|B) = \gamma B$. Define $D = A/B$. Using the law of iterated expectations it can be shown that $\mathbf{E}(D) = \gamma$. Now ...
2
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0answers
47 views

Finding the maximum value of summation of $\sin$ series.

The question at hand is to find the maximum value of: $$ \lim_{n \to \infty}\sum_{k=1}^{n} \frac{\sin{kx}}{k^{3}} = \frac{\sin{x}}{1^{3}} + \frac{\sin{2x}}{2^{3}} + \frac{\sin{3x}}{3^{3}} ............
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27 views

Meaning of Derivative of Summation

I wanted to ask something that I wasn't able to find online, and certainly haven't come across in my undergrad studies. Recently, I was solving a problem which had a function of the form $$ S(n) = \...