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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.

0
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3answers
19 views

How to calculate the number of elements in all subsets of a set

There are $2^N$ subsets of a set. For instance the set $\{ 1, 2 \}$ has the following subsets: $\{ \}$ $\{ 2 \}$ $\{ 1 \}$ $\{ 1, 2 \}$ I'm trying to calculate the total number of elements in all of ...
1
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2answers
44 views

General formula for the value of the $n$th derivative at $x=0$

Can anyone show me how to derive this summation for the value of the $n$th derivative at $x=0$ for this function: $\frac{d}{dx^n}(\exp({\frac{x^2}{2}+x}))$ is this sum: $\frac{d}{dx^n}(\exp({\frac{...
3
votes
2answers
48 views

Proving binomial sum equals $0$

My hypothesis is that, when $n \equiv 0 \mod 6$, then $$\sum_{k=0}^{n/3-1} \Bigg( \binom{n-2}{3k+1} 2^{3k+1}-\binom{n-2}{3k-1}2^{3k-1} \Bigg) = 0, \quad \binom{n-2}{3k-1} = 0 \text{ when } k=0$$ But ...
0
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1answer
41 views

What is the sum of the series? $\sum_{i=0}^{K+N-3}\left\lfloor \frac{i}{N-1} \right\rfloor$

What is the sum of the series given by $$\sum_{i=0}^{K+N-3}\left\lfloor \frac{i}{N-1} \right\rfloor$$ Where $N \in [2,100000)$ and $k \in [1,100000)$ Please help me find the first term and last term ...
-2
votes
1answer
70 views

Find the Value of $\sin^2A + \sin^2B + \sin^2C$ given the following data. [duplicate]

If $2\tan^2A\tan^2B\tan^2C + \tan^2A\tan^2B + \tan^2B\tan^2C + \tan^2C\tan^2A = 1$, then find the value of $\sin^2A + \sin^2B + \sin^2C$. My attempt 1). I tried to multiply both sides by $\cos^2A\...
0
votes
0answers
18 views

Convergence of a square sequence [duplicate]

$a = (a_1,a_2,...,a_n,...)$ is a sequence of real numbers. Suppose that for any given sequence of real numbers $b = (b_1,b_2,...,b_n,...)$ with $\sum_{n=1}^{\infty}b_n^2<\infty$ we know $\sum_{n=1}^...
0
votes
2answers
23 views

Adjusting weights of a weighted sum so that the sum has a lower bound.

Suppose I have this weighted sum (there are many $x_i$ sequences, so, I actually have many such sums): $$ S = \sum_{i=0}^n{x_i \times w_i} $$ With: $$ \sum_{i=0}^n{w_i} = 1 $$ Now, let's ...
5
votes
0answers
40 views

Sum involving roots of unity: $\sum \frac{1}{1-\omega^k}$

Let $\omega_n^k=\exp(2\pi ik/n)$ be the $n$-th roots of unity. I've come across the following sum a couple of times now (for example in a problem on hydrodynamics): $$ \sum_{k=1}^{n-1}\frac{1}{1-\...
0
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1answer
52 views

Is it true that $\sum_{i=1}^{n-1} \frac{1}{f(i)*f(i+1)}=\frac{n-1}{f(1)*f(n)}$ if $f(i)$,$f(i+1)$,$f(i+2)$… forms an arithmetic progression?

Well, '*' stands for a binary operation. I am asking this question because I derived the following 2 results: If $a_{i}$'s form an arithmetic progression then, $$\sum_{i=1}^{n-1} \frac{1}{\sqrt{a_{i+...
0
votes
1answer
23 views

How to get the sum till Kth term of this series for the given N ? I want the formula to get the sum till K'th term in this series

Here I have a series for N=2 -> 1,2,3,4,5,6,7,8,9,10 and the series continues for N=3 -> 0,1,1,2,2,3,3,4,4,5,5,6,6 and the series continues for N=4 -> 0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5 and the series ...
2
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2answers
65 views

How to find sum $\sum_{x=1}^K x^n(x-1)^m$

I want to find sum of this expression $\sum_{x=1}^K x^n(x-1)^m$ where $n$ and $m$ are constants around $10^5$ and $K$ is a constant around $10^9$. So my question is, is there a way to convert this ...
1
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0answers
32 views

Sum involving Stirling number of first kind

I am trying to solve https://www.rosecode.net/problem-550-A-specific-sum-over-all-subsets-of-a-set-askyear-2019 I shall quote the problem here. Given two natural numbers $N$ and $M$ with $0 < N \...
4
votes
1answer
77 views

If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$

If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\displaystyle \sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$. My attempt was to use firstly AM-GM in the denominator, like $a^3+5 \geq 3a+3$ and the ...
0
votes
2answers
48 views

The limit $n \rightarrow \infty$ of the standard deviation of $x_k= \ln k, k=1,2,3,..,n.$

The standard deviation for a sequence $x_k$ is defined as $$S_n=\sqrt{\sum_{k=1}^{n} \frac{{x_k}^2}{n}-\left(\sum_{k=1}^{n} \frac{x_k}{n}\right)^2}$$ By numerics the asymptotic value of $S_n$ for $x_k=...
0
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1answer
30 views

Sum of Binomial Related Series

I'd like to ask what is the closed form of $\sum_{i=a}^{n}\binom{i}{a}\cdot b^i$, given constant $a$, $b$ and $n$ with $a \leq n$? For $b=1$, the above equals $\binom{n+1}{n-a}$.
0
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0answers
22 views

Generate random vibration with zero mean for the motion of a point in 3D space

I want to simulate a point that moves with random vibration around a mean position (let's say around the position $[X, Y, Z] = [0,0,0]$). The first solution that I found is to sum a couple of ...
2
votes
2answers
48 views

Calculate the maximum value of $\sum_{cyc}\frac{1}{\sqrt{a^2 + b^2}}$ where $a, b, c > 0$ and $abc = a + b + c + 2$.

$a$, $b$ and $c$ are positives such that $abc = a + b + c + 2$. Caculate the maximum value of $$\large \frac{1}{\sqrt{a^2 + b^2}} + \frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}}$$ This ...
0
votes
2answers
63 views

How to find $\lim_{n \to \infty}\Sigma_{i=1}^n\frac{\sin(\pi x)^2}{i^2}$? [on hold]

I am trying to find: $$\lim_{n \to \infty}\sum_{i=1}^n\cfrac{\sin(\pi x)^2}{i^2}$$ using maple command: limit(Sum(sin(Pi*x)^2/i^2, i = 1 .. n), n = infinity); ...
0
votes
1answer
19 views

Exact Hat Algebra

When computing changes in $\lambda_{ij}$ defined as $\hat \lambda_{ij} = {\lambda_{ij}^{'} \over \lambda_{ij}}$, where does the $\lambda_{lj}$ in the sum in the denominator come from? $$\lambda_{ij} =...
1
vote
1answer
40 views

Find $ \cot\left(\sum_{n=1}^{23} \cot^{-1} (1+\sum_{k=1}^n 2k )\right)$

Find the value of $$ \cot\left(\sum_{n=1}^{23} \cot^{-1} (1+\sum_{k=1}^n 2k )\right)$$ I tried to simplify it but it became messy. How can I find $\sum_{n=1}^{23} \cot^{-1} (1+n+n^2)$? Please help ...
0
votes
2answers
28 views

Rewrite $\sum_{n=1}^k\log_3{(\frac{n+1}{n})}$ and write the formula in terms of k

Rewrite $\sum_{n=1}^k\log_3{(\frac{n+1}{n})}$ and write the formula in terms of k. I rewrote to $1+\frac{1}{n}$ and summed to get (I think) $\log_3(k+\frac{1}{n^k+k!})$ but I'm unsure if the $\log_3$ ...
3
votes
1answer
95 views
+50

Summation closed form

I have the following sum, $$\sum_{j=0}^{\lfloor\frac{i+n-1}{n+2}\rfloor}(-1)^{j}\binom{n}{j}\binom{i+n-j(n+2)-1}{n-1}+\sum_{j=0}^{\lfloor\frac{i+n-2}{n+2}\rfloor}2(-1)^{j}\binom{n}{j}\binom{i+n-j(n+2)...
1
vote
2answers
38 views

Rewrite $\sum_{n=1}^k{(n-1)/n!}$ and write the formula in terms of k

Rewrite $\sum_{n=1}^k{\frac{n-1}{n!}}$ I have turned it into $\frac{1}{n}*\frac{1}{(n-2)!}$ but do not know where to go from here.
0
votes
1answer
22 views

Product of Two Summations with Different Upper Limits

I'm trying to multiply two different finite summations with different upper limits. I've tried Cauchy Product but i think it's valid for same upper limits. I also tried to split the summation. Any ...
2
votes
3answers
73 views

Calculate the limit including binomial coefficients

$\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$ It's clear that we can simplify the limit a little bit, after which we get: $\lim\limits_{n\to\...
0
votes
1answer
26 views

Can Leibniz integral rule be extended to differentiation under the sigma sign?

To differentiate $\displaystyle M(t)=\sum_i e^{tx_i} P(x_i)$ with respect to $t$, for instance.
0
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1answer
16 views

Inequality relations used in Multiple Sums

I am reading Chapter 2 of Concrete Mathematics, and have some trouble understanding the rationale behind two identities used for simplification. ...
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0answers
22 views

How to show that if $\sum_{d|(p-1)} \psi(d) = \sum_{d|(p-1)} \phi (d)$ then $\psi(d) = \phi (d)$

The argument used in the book is that they will show that $\psi (d) \leq \phi(d)$ for each divisor $d$ of $p-1$ because this in conjunction with the equality of the two sums in the title will produce ...
0
votes
0answers
22 views

Solve and asymptotic expansion of $\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$

I am solving constrained polynomial systems resulting in constrained sums. I am looking to see if $$\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$$ is expressible in ...
1
vote
6answers
231 views

Prove that $\sum\limits_{x=0}^\infty \frac{1}{(x+ 1)(x+2)} = 1$.

Prove $$\sum_{x=0}^\infty \frac{1}{(x+ 1)(x+2)} = 1.$$ I couldn't find this problem solved online and I haven't reviewed series in a long time. I thought maybe squeeze theorem could help? A related ...
1
vote
0answers
38 views

Divisibility of formula

I have a formula which i think must be divisible by prime numbers, but I cannot seem to prove it. $$f(n) = \frac{3^n -\displaystyle \sum_{k=0}^{\left \lfloor{\frac{n}{3}}\right \rfloor } \binom{n}{3k}...
0
votes
3answers
29 views

Can't understand an equality between sums

During an induction proof I came across an equality that I can't understand. During the last step of the induction there is: $$ \sum_{k=1}^{n}{(k+1){n \choose k-1}} = \sum_{k=0}^{n-1}{(k+2){n \choose ...
0
votes
2answers
69 views

Is there such an infinite sequence, such that $\lim_{n\to\infty} \frac{\sum_{i=1}^{n}a_n}{2n}=\text{ exact form constant}?$

I will try to ask my question as clear as possible. We know that, there exist infinitely number of infinite sequences that, consist of elements $\left\{0,1,2 \right\}$, which is can not express by ...
0
votes
2answers
55 views

Prove that $\sum_{k=0}^{2n} \frac{1}{\pi^n+\pi^k}={\frac{2n+1}{2\pi^n}}.$

Prove that $$\sum_{k=0}^{2n} \frac{1}{\pi^n+\pi^k}={\frac{2n+1}{2\pi^n}}.$$I tried telescoping taking $$ t_k=\frac{1}{\pi^n+\pi^k}$$ but I am unable get the differencing: $t_k=f_k-f_{k-1}.$ Can some ...
1
vote
1answer
44 views

Solving $\begin{cases} T_{0}=5 \\ 2T_{n}=nT_{n-1} + 3n! \end{cases}$ using method of summation factors

I need to solve the following recurrence relation using "method of summation factors". $\begin{cases} T_{0}=5 \quad \quad \quad \quad \quad\quad\quad \iff n=0\\ 2T_{n}=nT_{n-1} + 3n! \quad \quad \iff ...
0
votes
1answer
35 views

How to calculate $\sum_i^m \sum_j^n x_i x_j (\phi_i^T \phi_j)$?

I'm wondering how to tackle down and calculate the following equation efficiently: $\sum_{i=1}^m \sum_{j=m+1}^n x_i x_j (\phi_i^T \phi_j)$ Am I allowed to do this? $(\sum_i^m x_i) (\sum_j^n x_j) (\...
0
votes
2answers
41 views

General formula of partial sum of series (non-geometric)

Like user71317 in his question I am struggling to understand how we arrive at the general formula of the partial sums of a series. In my case the following series: $$\sum_{n=2}^{\infty} \frac{1}{n^2-...
2
votes
4answers
67 views

Evaluating $\sum_{k=1}^{n} \frac{1}{k(k+1)}$

I have just started learning sums. I need to evaluate the following sum: $$S_{n} = \sum_{k=1}^{n} \frac{1}{k(k+1)}$$ $$a_{1} = \frac{1}{2}, a_{2} = \frac{1}{6}, a_{3} = \frac{1}{12}, a_{4} = \frac{1}...
0
votes
1answer
25 views

Compute difficult sum

How can I compute this sum? $\sum_{k=0}^\infty\left[\frac{(2k+1)!}{(2k+b)}-\frac{1}{(2k)!(2k+b)}-\frac{(2k)! }{(2k+b+1)}+\frac{1}{(2k+1)!(2k+b+1)}\right]$ I see the $k=0$ term is equal to zero, but ...
2
votes
1answer
47 views

Prove that $\sum_{n=1}^{\infty}\log \cos \left (\frac{1}{n}\right )$ converges absolutely.

Prove that $$\sum_{n=1}^{\infty}\log \cos \left (\frac{1}{n}\right )$$ converges absolutely. The answer here suggests to use the Limit Comparison Test but it works for $a_n \geq 0$ while $\ln(\cos (1/...
0
votes
2answers
71 views

Proving integral $\int_0^1\frac{e^x-1}{x}$ is equal to $\sum_{n=1}^{\infty}\frac{1}{n \cdot n!}$

Show that the following equality is true. $$ \int_0^1\frac{e^x-1}{x}\, \mathrm dx = \sum_{n=1}^{\infty}\frac{1}{n \cdot n!} $$ How can I tackle this problem?
1
vote
2answers
51 views

Closed form of many repeated summations of n

I was looking into double summations, then I thought of repeated summations. As of now, I am having difficulty simplifying, for example $$\sum_{r=1}^8...\sum_{z=1}^y\sum_{n=1}^z\sum_{i=1}^n i$$ ...
1
vote
1answer
36 views

Prove that $\sum_{k=0}^{\infty}{\frac{(3i)^k}{k!}}$ converges .

by the ratio test:$$\bigg |\frac{a_{k+1}}{a_k}\bigg |=\frac{\frac{(3i)^{k+1}}{(k+1)!}}{\frac{(3i)^k}{k!}}=\frac{3i\cdot k!}{(k+1)!}=\frac{3i\cdot k!}{k!\cdot (k+1)}=\frac{3i}{k+1}\overset{k\to \infty} ...
0
votes
1answer
39 views

Snails and Sums

At the beginning of a $10\,\mathrm m$ long rubber band sits a snail. Every day it crawls one meter ahead. Every night, when the snail is resting, an evil man stretches the tape evenly by $10\,\mathrm ...
0
votes
1answer
21 views

Equating coefficients of like terms involving a summation

I am working through this problem, and looking at the answers provided below, I can't seem to understand how the answers are arrived at. $$u(x,1)=A_0+\sum_{n=1}^\infty A_n\sinh(n)\cos(nx)=1+\cos(2x)...
0
votes
3answers
36 views

How to work out sum of Square Number from given numbers?

Which of the following cannot be written as the sum of two distinct square numbers? A.106 B. 109 C. 112 D. 117 What would be the correct answer here and can someone explain in detail please.
0
votes
2answers
47 views

A die is thrown $k$ times. How to determine the sum equal to $k$?

A die is thrown $k$ times. Compute the probability of these resulting sums: $k$ $k+1$ In this problem I assume that the die is $6$-sided and that the answer should be in terms of $k$. Formula: $p(...
0
votes
0answers
57 views

Integral and Summation relationship [closed]

Summation $$\sum_{k=-\infty}^{+\infty}x_k$$ isn't discrete version concept of Integrals $$\int_{-\infty}^{+\infty} f(x) dx$$, that is obvious. Nevertheless I don't understand why when we come from ...
2
votes
2answers
72 views

Find the value of $a_4-a_2$

Let $a_1<a_2<a_3<a_4$ be positive integers such that $\sum_{i=1}^{4}\frac{1}{a_i}=\frac{11}{6}$. Find the value of $a_4-a_2.$ I do not know how to proceed. I have tried to simplify the ...
0
votes
1answer
39 views

What is $2\sum^2_{i=1}\sum^2_{j>i}a_{ij}x_ix_j$ is equal to?

Intuitively I did: $$2\sum^2_{i=1}\sum^2_{j>i}a_{ij}x_ix_j=2(a_{12}x_1x_2)+2(a_{22}x_2x_2)$$ But I'm not sure, since I'm new to calculus.