Questions tagged [summation]

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

0
votes
0answers
13 views

Summation explanation

I'm wondering how the following summation could be explained: $\sum_{1 \leq i < j \leq K} (X_{i} - X_{j})^3$ Is this equivalent to $\sum_{1 \leq i \leq K} \sum_{i < j \leq K}(X_{i} - ...
1
vote
2answers
30 views

Prove that inequality $27+7\left(ab+bc+ca\right)\le 8\left(\sum \sqrt{a+3b}\right)$

For the positive reals $a,b$ and $c$ so that $a+b+c=3$. Show that $$27+7\left(ab+bc+ca\right)\le 8\left(\sqrt{a+3b}+\sqrt{b+3c}+\sqrt{c+3a}\right)$$ That inequality has been created by Imad Zak, i ...
0
votes
0answers
15 views

simplify the summation of harmonic cosines

Is it possible to simplify the following summation of harmonic cosines eliminating the sum? \begin{equation} \sum_{n=1}^{N-1} \left( 1 + \cos \left( \frac{\pi n}{N} \right) \right)^N \end{equation} ...
0
votes
1answer
19 views

How to prove the interchangability of sums using monotone convergence theorem?

Let $a:\mathbb{N^2}\rightarrow\mathbb{R}^+$. Show that $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} a(n,m) =\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} a(n,m) $$ using the monotone convergence theorem. So ...
6
votes
5answers
416 views

Proving inequality for induction proof

In an induction proof for $\sum_{k = 1}^n \frac{1}{k^2} < 2 - \frac{1}{n}$ (for $n \geq 1$), I was required to prove the inequality $\frac{1}{(n+1)^2} + \frac{1}{n+1} < \frac{1}{n}$ This is my ...
-1
votes
1answer
113 views

Proof of $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ [duplicate]

Prove that $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ I have actually got the HINT for this Proof from a very nice book: The HINT i started with is: $$\frac{1}{\sin^2 x}=\frac{1}{4\sin^2\...
3
votes
1answer
86 views

Computing $\sum_{k=1}^n (a^k \bmod m)$

I would like to find a closed form solution for $$\sum_{k=1}^n (a^k \bmod m)$$ $$0<a<m, n > 0$$ Note that the mod operator is within the brackets. If a closed form solution does not exist, ...
1
vote
2answers
39 views

How to calculate sum of floor functions.

Let $f(x)$ is real valued function. How to calulate or at least find lower and upper bound of sum: $$\sum_{1\le n \le x}\left\lfloor f(n) \right\rfloor$$? For example when $f(n)=\frac{x}{n}$ ...
0
votes
0answers
27 views

Summation of Radical n

How can I calculate this Summation based on $n$ ? $$S = \sum_{k=1}^\infty\sqrt[2^k]{n}$$
1
vote
1answer
32 views

Probability mass function of sum of random variables

Let $\mathbb P(L=k) = \frac{140}{223(k+2)}, k = 0, 1, ..., 5.$ Credit portfolio consists of 8 independent loans, where $L_i, i = 1, ..., 8$ is the number of “physically” possible defaults of one loan....
0
votes
0answers
14 views

Probability mass function of sum of independent random variables $L_i$

Let the credit portfolio consists of $5$ independent loans, where $L_i, i = 1, ..., 5$ is the number of “physically” possible defaults of one loan. Each of $L_i$ ~ $B(1,\frac{1}{3Z+1}),$ $i= 1,...,5....
3
votes
2answers
182 views

What is integer part of $\sum_{j=2}^{100}\frac{1}{\sqrt{j}}$? [closed]

what is integer part of $$\sum_{j=2}^{100} \frac{1}{\sqrt{j}}= \frac{1}{\sqrt2}+\frac{1}{\sqrt3}+\frac{1}{\sqrt4}+\cdots+\frac{1}{\sqrt{100}}$$ I don’t have any idea how to approach this problem.
0
votes
1answer
33 views

$\sum_\eta \exp(b*(\eta_1 + …+\eta_n)) = (1+e^b)^n$.

I'm wondering why this is a correct relation, $\eta$ is the collection of $\{\eta_1,...,\eta_n \}$; ${\eta_1,...,\eta_n }$ are independent and the given relation is : $\sum_\eta \exp(b*(\eta_1 + ...+\...
0
votes
2answers
59 views

Evaluating $\displaystyle\sum_{k=1}^{n-1} k\cos (k x)$

My intent is to find an expression for the following summation, $$\sum_{k=1}^{n-1} k\cos (k x).$$ We know that (cfr. Sines and Cosines of Angles in Arithmetic Progression by M. P. Knapp) $$\sum_{k=0}^{...
1
vote
2answers
62 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{...
-2
votes
3answers
70 views

Convergence of a double series $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{(m+n)^2}$ [duplicate]

Does the series $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{(m+n)^2}$ converge?
5
votes
6answers
337 views

Inequality. $\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}+\frac{c}{\sqrt{a}}\geq3$

Let $a,b,c \in (0, \infty)$, with $a+b+c=3$. How can I prove that: $$\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}+\frac{c}{\sqrt{a}}\geq3 ?$$. I try to use Cauchy-Schwarz rewriting the inequality like : ...
1
vote
2answers
40 views

$\sum_{k=1}^{100}k\cdot 2^k\geq 99\cdot 2^{101}$ solve geometrically?

I've succeeded proving it using induction, but I wondered if there exists a prettier solution, maybe prooving it geometrically with areas of rectangles or some other way.
5
votes
0answers
109 views

An integration-via-summation formula

For symbolic transformation of integrals and series I occasionally use this formula: $$\int_0^1f(x)\,dx=-\sum_{n=1}^\infty\sum_{m=1}^{2^n-1}\frac{(-1)^m}{2^n}f\left(\frac m{2^n}\right)\tag{$\diamond$}$...
1
vote
3answers
37 views

Alternative formula for number of diagonals in a polygon?

While teaching a secondary school student stochastic I found that the sum $$\sum\limits_{n=1}^N(n-2)$$ is equal to the formula used to calculate the number of diagonals in a polygon with N sides: $$\...
2
votes
3answers
80 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
3answers
25 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 ...
3
votes
2answers
59 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
votes
1answer
47 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 ...
0
votes
0answers
53 views

Renewal Processes (Rolling a Fair Die)

What is the expected number of rolls until you see a 6 immediately followed by a 2? Let $D_i$ = number observed on the $i^{th}$ roll Assume $T_0=0$ Now let $T_1$ = min{$i\ge2$:$D_{i-1}=6$, $D_i=2$}...
-2
votes
1answer
71 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
19 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}^...
3
votes
2answers
83 views

Distribution of $\sum_k \alpha^k X_k$

Let $(X_k)$ be a sequence of independent Bernoulli random variables, such that $\Pr[X_k = 1] = p$. Then for $0\le\alpha<1$ the sum $$\sum_{k=0}^\infty \alpha^k X_k$$ is real random variable in the ...
4
votes
1answer
149 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)...
0
votes
1answer
56 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
2answers
24 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 ...
2
votes
2answers
69 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 ...
5
votes
0answers
43 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-\...
14
votes
2answers
457 views

Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$

I realize that there is probably not a closed form, but is there an efficient way to calculate the following expression? $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$ I've noticed $$\sum_{k=...
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 ...
4
votes
1answer
79 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
votes
2answers
65 views

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

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); ...
1
vote
0answers
36 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 \...
0
votes
1answer
31 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
votes
1answer
86 views

Expansion of the cube of the sum of N numbers

I know that the expansion of the square of a summation can be expressed as: $$\left( \sum_{n=1}^N a_n \right)^2 = \sum_{n=1}^N a_n^2 + 2 \sum_{j=1}^{N}\sum_{i=1}^{j-1} a_i a_j$$ where $a_n \in \...
0
votes
0answers
25 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
50 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
1answer
21 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
41 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
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 ...
0
votes
2answers
30 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 $\log_3$ ...
0
votes
2answers
41 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.
15
votes
0answers
629 views

Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$ [duplicate]

A teacher gave this as a homework question, and I have tried but haven't been able to arrive at a solution. $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$ Could someone prove it, or at least ...
1
vote
6answers
234 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 ...