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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.

2
votes
3answers
48 views

Maximizing the sum $\displaystyle \sum_{n=1}^m \sin n$

Consider the sum : $$\displaystyle \sum_{n=1}^m \sin n$$ For which value of $m,$ we will obtain the maximum sum? Here's my approach : $\displaystyle \sum_{n=1}^m \sin n=\dfrac{\sin 1}{4 \sin^2 \dfrac{...
0
votes
0answers
25 views

WHat does double notation in the given formula means in the given formula

Hi I am trying to understand how to do computation for the given double summation . Please find pic attached the entire formula . Please help me to understand ?? How can i solve this in excel ? shall ...
3
votes
0answers
27 views

Examples where we cannot interchange summation and integration

We know of conditions under which we can interchange summation and integration (1, 2). What are some simple examples where we cannot do so and which we could present to high-school/introductory ...
2
votes
3answers
34 views

Why is $ \sum_{k = 1}^{n - 1}O\left( \binom{n}{k}k^2 \right) = O(n^22^n)$?

In the analysis of an exact dynamic programming analysis for the Travelling Salesman problem in Exact Exponential Algorithms by Fomin and Kratsch, it is stated on p. 6 that $$ \sum_{k = 1}^{n - 1}O\...
0
votes
1answer
27 views

Summation from a real number to a complex number

What would the solution of this be: $\displaystyle \sum_{n=1}^i n$ where $i$ is the imaginary unit. or any other formula really, I'm just interested to know how would a summation work from a real ...
0
votes
1answer
25 views

Sum of m dice rolled n times where sum of each dice is lower than some value

Let there be $m$ dice (not neccessarily all same-sided, but even when they are I don't have a solution). Each dice is rolled $n$ times. The sum of $n$ rolls of all dice needs to be $T$. The sum $X_i$ ...
0
votes
1answer
42 views

Basic Maths…can’t get my head around this.

This might be very basic but... Take any number for example 2356 now 2+3+5+6=16 and 1+6=7 or 236+5=241 and 2+4+1=7 and also 21+4=25 again 2+5=7 or 652+3=655 which digits sum is 7 again and then 65+5=...
0
votes
1answer
53 views

Simplify $x^n + x^{n-1} + … + x^1 + 1$ [duplicate]

Can I somehow simplify $x^{n-1}+x^{n-2} + ... + x^{2} + x^{1} + 1$? I would like to have an explicit formula for that sum, but could not figure out a way to do so. Could you help me? Thanks!
1
vote
4answers
55 views

How to apply induction to this formula?

I want to justificate following equation: $$\sum_{k=0}^n \frac{(-1)^k}{k!(n-k)!}\frac{1}{2k+1} = \frac{2^n}{(2n+1)!!}$$ I calculated the both sides for $n$ from 1 to 10 and it was true. How the ...
0
votes
0answers
17 views

Closed form for a series involving binomial coefficients

I am unable to find a closed form for $\sum\limits_{r=1}^{min(n-k+1, k)} {2^r}{{n-k}\choose r-1}{{n-k+1}\choose r}$. Any help would be appreciated.
1
vote
0answers
38 views

Evaluate $\sum_{m=1}^{\infty} \frac{1}{2m+s}\frac{1}{2m+r}$ [duplicate]

$S = \sum_{m=1}^{\infty} \frac{1}{2m+s}\frac{1}{2m+r}$ $r>s>0$ and both $r,s$ integers. write $x=2m+s$ and $y=r-s$. \begin{align} &S = \sum_{m=1}^{\infty} \frac{1}{2m+s}\frac{1}{2m+r} = \...
0
votes
3answers
27 views

Summing a Sequence

I recently came across a problem that proved particularly challenging. Unfortunately, I have not been able to complete the problem despite multiple different methods tried. It is as below: Show that ...
0
votes
2answers
39 views

Matrix and two vectors product

Does anyone know how to represent the product of an $n×n$ matrix and two $n$-vectors in a compact form using $\sum$ and indexes? If it was only a vector then $\sum_{j=1}^{n} A_{i,j}v_j$. But with two ...
1
vote
1answer
23 views

How Can I Represent These Progressions in Sigma Notation?

I would like to represent the following finite progressions in sigma notation: $Finding\ the \ n^{th} \ term \ of \ a \ geometric \ progression$: $a_n=a_1(r^{n-1})$, where $a_1$ is the first time and ...
1
vote
1answer
15 views

Solving for mixture coefficients in Gaussian Mixture Model

In Chapter 9 of 'Pattern Recognition and Machine Learning' Bishop explains the Expectation-Maximization algorithm, also with the application on Gaussian mixtures. On page 436 (14 in the link) he ...
-1
votes
1answer
97 views

Show that $\frac {1}{3+x^2+y^2} + \frac {1}{3+y^2+z^2} +\frac{1} {3+x^2+z^2}\leq \frac {3}{5} . $

Let $x, y, z>0$ s.t. $x+y+z=3$. Show that $$\frac {1}{3+x^2+y^2} + \frac {1}{3+y^2+z^2} +\frac{1 } {3+x^2+z^2}\leq \frac {3}{5}\ . $$ My idea: $$3 + x^2 + y^2 \geq 1 + 2x+ 2y=7-2z $$ I notice ...
2
votes
2answers
55 views

$\sum\limits_{k=1}^nk^r = \frac{1}{1+r}n^{r+1}+a_rn^r +…+a_1n$

Show that for every $r \in \mathbb{N}$ there are numbers $a_1,..,a_r \in \mathbb{Q}$ such that $\sum\limits_{k=1}^nk^r = \frac{1}{1+r}n^{r+1}+a_rn^r +....+a_1n$ for all $n \in \mathbb{N}$ There ...
1
vote
1answer
22 views

Finding the geometric sum of this recurrence

I'm having trouble with evaluating geometric sequences that look like this: $Cn\sum_{i=0}^{\log_3n} (5/3)^i$ where $n$ is the number of operations, and $Cn$ just represents $n$ times some constant $...
1
vote
1answer
41 views

Calculate $\sum_{n\ge2}\log\left(1-\frac1{n^2}\right)$

This is the expression whose sum I have to calculate: $$\sum_{n\ge2}\log\left(1-\frac1{n^2}\right)$$ I have tried to use the mengoli's series properties but I failed. The listed answer should be $-...
3
votes
4answers
71 views

Computing the value of $\frac{1}{3^2+1} + \frac{1}{4^2+2} + \frac{1}{5^2+3}…\infty$=?

I have tried converting this series into a telescopic sum whose terms could cancel out but haven't succeeded in that effort. How should I proceed further?
0
votes
0answers
38 views

What is the general term of $\sum_{i=2}^{n/2} (n-i)!$?

How do you go about solving this summation?: $\sum_{i=2}^{n/2} (n-i)!$ Edit: For context, I am trying to do an analysis of the average case time complexity of merge sort, measured based on the ...
2
votes
2answers
40 views

Infinite summation question and i need to make an algorithm for finding the summation?

$$\sum_{n=1}^{\infty} \frac {x^{2n -1}} {2n!}$$ for the algorithm i use == $d=\frac {a_{n}} {a_n - 1}$ And other hint that I have is the for $\sum_{n=1}^{\infty} \frac {x^{2n}} {n!}$ ; $d = \frac{x}...
3
votes
2answers
39 views

A proof on multinomial roots

If $x_1,x_2,...,x_{n-1},x_n$ be the roots of the equation $$1 + x + x^2 + ... + x^n = 0$$ and $y_1,y_2,...,y_{n},y_{n+1}$ be those of equation $$1 + x + x^2 + ... + x^{n+1} = 0$$ show that $$(1-x_1)(1-...
8
votes
0answers
94 views

Prove a strong inequality $\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2-\frac{7\ln 2}{8\ln n}\right)\sum_{k=1}^n\frac 1{a_k}$

For $a_i>0$ ($i=1,2,\dots,n$), $n\ge 3$, prove that $$\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2\color{red}{-\frac{7\ln 2}{8\ln n}}\right)\sum_{k=1}^n\frac 1{a_k}.$$ The case without $\...
0
votes
1answer
50 views

show that $\frac {a_{N+1}}{s_{N+1}} +…+\frac {a_{N+k}}{s_{N+k}}\geq 1 -\frac {a_{N}}{s_{N+k}} $ [duplicate]

Suggestion of how to do it, please. Suppose $\{a_n\}$ is a succession in $\mathbb R ^+$ such that $\sum a_n$ diverges, and if $s_n = \sum\limits_{k=1}^n{a_k}$. show that $\frac {a_{N+1}}{s_{N+...
0
votes
0answers
39 views

For integer $n>2$, and real $k$, is it true that $\sum_{i=0}^{n-1} \cos\left( k+\frac{4i\pi}{n}\right )=0$?

Let $n$ be an integer, $n>2$ and $k$ be a real number. Is the following true? $$\sum_{i=0}^{n-1} \cos\left( k+\frac{4i\pi}{n}\right )=0$$ Which identity should I use? How can I solve it? Please ...
0
votes
1answer
30 views

What is the limit of sequence with sum? [duplicate]

We have N numbers $x_{k} \geq0 $. What is the $\lim _ { n \rightarrow \infty } \sqrt [ n ] { x _ { 1 } ^ { n } + \ldots + x _ { N } ^ { n } }$ ? Any hints?
0
votes
2answers
18 views

renormalization of sum of continuous random variables

I want to sum two random variables. So $Z = X+Y$ and $f_{X+Y}(z)=\int_{-\infty}^{\infty}f_{xy}(x,z-x)dx $ So I wanted to test this out a bit. If $$f_{xy}(x,y) = 6∙10^{-4}(x^2+y^2)$$ when $-5\leq x ...
-4
votes
1answer
31 views

Why following the below conditions, the $\sum_{i=0}^{\infty} a_n*b_n$ in converge? [closed]

Given $a_n$ a positive series and $\sum_{i=0}^{\infty} a_n$ is converge, and given also that $\lim_{n\to \infty} b_n = 100$, then the summation $\sum_{i=0}^{\infty} a_n*b_n$ is converge.
1
vote
0answers
31 views

Show that $\sum_{l=0}^{\infty}(\gamma\matrix{A})^l$ converges

Based on the following conditions where $\matrix{A}$ is a matrix: $\left| \lambda\matrix{A} \right|=\left| \lambda \right|\left| \matrix{A} \right| $ for any $\lambda \in \mathbb{R}$ $\left| \matrix{...
2
votes
0answers
39 views

Simplifying Likelihood Ratio

I have a data set $((Y_1,x_1),(Y_2,x_2),...,(Y_n,x_n))$ where $Y_i$ is distributed as $N(\theta x_i,1)$. I want to perform a likelihood ratio test for $\theta$ to investigate the hypothesis $H_0: \...
2
votes
2answers
64 views

show that $\frac{1}{3n+1}+\frac{1}{3n+2}+…+\frac{1}{5n}+\frac{1}{5n+1} < \frac{2}{3}$ , $\forall n \mathbb \in{N}$ [closed]

Show that $$\frac{1}{3n+1}+\frac{1}{3n+2}+...+\frac{1}{5n}+\frac{1}{5n+1} < \frac{2}{3}$$ for all $n \in \Bbb{N}$ I tried with induction method but I can not find any results.
0
votes
1answer
17 views

How do you notate recursive summation?

I have devised the following expression and was just wondering if there is some way to better notate it mathematically. $$ \sum_{i=0}^{n} \Bigg( \sum_{j=i}^{n} \bigg( \sum_{k=j}^{n} \cdots \sum_{z=y}^...
-2
votes
1answer
88 views

Evaluating the sum of $\sum_{k=0}^n \binom{n}{k}^2$ [duplicate]

I don't know if this question is trivial but let me put it in the first place. I'm trying to find the sum of $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n-1}^2+\binom{n}{n}^2$ or ...
1
vote
3answers
36 views

How to work with sums? $\sum$

During high school I missed out on some of the math lessons due to health issues, one thing I missed out on was working with sums. I am now a second-year engineer student and I am still amazed how bad ...
6
votes
1answer
66 views

Find $\sum_{n=1}^\infty\frac{2^{f(n)}+2^{-f(n)}}{2^n}$, where $f(n)=\left[\sqrt n +\frac 12\right]$ denotes greatest integer function

Question: Let $f(n)=\left[\sqrt n +\dfrac 12\right]$, where $[\cdot]$ denotes greatest integer function, $\forall n\in \Bbb N$. Then, $$\sum_{n=1}^\infty\frac{2^{f(n)}+2^{-f(n)}}{2^n}={\color{red}?...
1
vote
2answers
68 views

Why the $\lim_{n\to\infty} (\frac{n}{n^2+1}+\frac{n}{n^2+4}+…+\frac{n}{n^2+n^2})= \frac{\pi}{4}$?

Why the $\lim_{n\to\infty} (\frac{n}{n^2+1}+\frac{n}{n^2+4}+.....+\frac{n}{n^2+n^2})= \frac{\pi}{4}$? I read somewhere that it is related to $f(x)=\frac{1}{1+x^2}$ but dont know why...
-1
votes
1answer
31 views

why this statement about $\sum_{i=0}^n a_n$ is false?

Given that ${a_n}$ is positive series, and $\sum_{i=0}^n a_n$ is converge: -There is a sub-series for ${a_n}$ that converge to S>$0$. (THIS statement is false) why is this statement false?
3
votes
1answer
48 views

Double sum of binomial coefficients quotients

How to prove the following formula? $$\sum_{j=0}^{n-1} \sum_{k=0}^{j} \frac{{n}\choose{k}}{{n-1}\choose{j}} = 2^{n-1}\sum_{j=0}^{n-1} \frac{1}{{n-1}\choose{j}}$$ I tried induction and manipulating ...
-1
votes
1answer
75 views

Let $a, b, c>0$ such that $abc=1$. Prove that $\sum{a(b^{2}-\sqrt{b})}\geq 0$. [closed]

Let $a, b, c>0$ such that $abc=1$. Prove that $\sum{a(b^{2}-\sqrt{b})}\geq 0$. (By $\sum$ I mean cyclic sum) My attempts include algebraic manipulation and many uses of AM-GM inequality, but I ...
0
votes
2answers
36 views

Derivatives Across Summations

So, I took one intro course in Tensor calculus and this problem reminds of that, except I can't quite recall how derivatives work with respect to components, or what those derivatives produce. ...
0
votes
0answers
39 views

Simplifying $\sum_{i=0}^{\log n} \frac{n}{\log\left(\frac{n}{2^i}\right)}$

$$\sum_{i=0}^{\log n} \frac{n}{\log\left(\frac{n}{2^i}\right)}$$ I'm having trouble seeing how this summation simplifies. It seems it would be something like: $$\frac{n}{\log(n)} + \frac{n}{\log\...
0
votes
0answers
35 views

How to plot an infinite series

I have this equation to define the velocity in a duct $$ u(y,z) = \frac{16 a^2}{\alpha \pi^3} k \sum_{i=1,3,5...}^{\infty} (-1)^{\frac{(i-1)}{2}} \left[ 1- \frac{\cosh(\frac{i \pi z}{2 a})}{\cosh (...
1
vote
2answers
51 views

When adding up two summations, is the “plus” sign wrong?

I have a fraction where in the denominator I need to sum two products (numbers and their weights). For this, I inserted two Sigmas but now I am unsure whether I can have Sigma + Sigma or just Sigma ...
0
votes
0answers
28 views

summation and Dirichlet product of two functions $\mu$ and $F$.

Let $f(x)$ be defined for all rational $x$ in $[0,1]$ and let $$F(n) = \sum_{k=1}^n f\left(\frac kn\right),\qquad F^*(n) =\sum_{\substack{k=1\\(k,n)=1}}^n f\left(\frac kn\right)$$ How to prove that $\...
0
votes
1answer
21 views

$\sum_{n>k}\frac{j}{j+1}\cdot (j+1)^{-n}=\frac{j}{j+1}\cdot (j+1)^{-(k+1)}\cdot \sum _{n=0}^{\infty}(j+1)^{-n}$

Given $1\leq j\leq 6$ and $k\in \mathbb{N}$, why is this equality holds? $\sum_{n>k}\frac{j}{j+1}\cdot (j+1)^{-n}=\frac{j}{j+1}\cdot (j+1)^{-(k+1)}\cdot \sum _{n=0}^{\infty}(j+1)^{-n}$
0
votes
0answers
16 views

Proving an inequality involving complex numbers.

Let $z \in \mathbb C$, and $n$ be a natural number $\ne 0.$ How would you prove that $$\left|\sum_{k=0}^n\frac{z^k}{k!}-\left( 1 +\frac{z}{n} \right)^n\right| \le \sum_{k=0}^n\frac{|z|^k}{k!}-\left( 1 ...
0
votes
0answers
19 views

A Summation of a Sequence of Nested Summations [closed]

Let $$f(n)=\sum_{a_1=1}^{9}{\left[ 10^{n-1}a_1 + \sum_{a_2=a_1}^{9}{\left[ 10^{n-2}a_2 + \sum_{a_3=a_2}^{9}{\left[ 10^{n-3}a_3+ \cdots + \sum_{a_n=a_{n-1}}^{9}{10^{n-n}a_n}\right]}\right]}\right]}$$ ...
3
votes
4answers
97 views

Integrate $\int_0^1{\frac{\ln{x}}{1+x}}dx$ using $\sum{\frac{1}{k^2}}=\frac{\pi ^2}{6}$

Integrate $$\int_0^1{\frac{\ln{x}}{1+x}}dx$$ using $$\sum{\frac{1}{k^2}}=\frac{\pi ^2}{6}$$ My Attempt: I know that $$\lim_{n\to{\infty}}\sum_{r=0}^nf\bigg(\frac{r}{n}\bigg)\cdot\frac{1}{n}=\int_0^...
-2
votes
2answers
62 views

How to solve $\sum_{k=1}^{n} a_{k} = \left(a_{n} + \frac{1}{4}\right)^{2}$? [closed]

How to solve $$\sum_{k=1}^{n} a_{k} = \left(a_{n} + \frac{1}{4}\right)^{2}$$ for any $n$ with $a_n > 0$ ? I don't know the method of solving this recurrence relation. Hints would be appreciated.