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

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3
votes
2answers
147 views

Euler-Maclaurin Summation

Using EM summation formula estimate $$ \sum_{k=1}^n \sqrt k $$ up to the term involving $\frac{1}{\sqrt n}$ My attempt is $$ \sum_{k=1}^n \sqrt k = \frac{2 \sqrt{n^3}}{3} -\frac{2}{3} + \frac 1 ...
0
votes
6answers
146 views

Find the sum $1^2+3^2+…+(2n-1)^2$ without induction

I tried with $$(2n-1)^3-(2n-2)^3=12n^2-18n+7$$ Now, forming partial sums for $n=1,2,...$ $$(1^3-0^3)+(3^3-2^3)+...+((2n-1)^3-(2n-2)^3)=12(1^2+...+n^2)-18(1+...+n)+7n$$ How to express ...
0
votes
3answers
61 views

Simplifying $\sum_{i=1}^n {1\over i(i+1)}$ to ${n\over n+1}$

I need to get from this: $$\sum_{i=1}^n {1\over i(i+1)}$$ to: $${n\over n+1}$$ or: $${1 - {1\over n+1}}$$ I have tried looking for sums identities with fractions, using WolframAlpha.com (that's how I ...
0
votes
1answer
38 views

Expanding a series with a constant [duplicate]

How do you expand the series: The sum, starting at j=1 and ending at 6 when the value next to the summation symbol on the right is 4? Are all of the terms of the series 4?
0
votes
1answer
48 views

Closed-form for rational power sum

$$s_n = \sum_{k=1}^n e^{1/k}$$ This sum came up while practicing closed-form finding on a calculus book's infinite series chapters. Using Concrete Mathematics' perturbation method, I arrive at ...
1
vote
0answers
37 views

transformation: Integration to summation

Good day! I'm studying right now some transformation and I encountered the following equation: $$(2\pi)^{-n/2} \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \exp\left(-\frac{1}{2} ...
0
votes
1answer
43 views

What is the limit of the ratio of the sum of all real numbers from 0 to 2 over the sum of all real numbers from 0 to 1.

I approached this question I made, by saying that the sum of a finite amount of numbers from a to b separated by a common change is average•(# of numbers we have). I found the formula the # of the ...
0
votes
2answers
58 views

Closed-form expression for $\log(x) + \log(x^2) + \cdots + \log(x^n)$

Does anyone know of a closed-form expression for the following? $$ \log(x) + \log(x^2) + \cdots + \log(x^n) $$
1
vote
3answers
70 views

Prove by induction: $\sum\limits_{i=1}^{n}(4i+1) = 2n^2 + 3n$

Prove by induction: $$\sum\limits_{i=1}^{n}(4i+1) = 2n^2 + 3n$$ It's just the numbers that confuse me; I know how to do a simple induction proof that first for $p(k)$ and then for $k+1$ etc but ...
3
votes
2answers
97 views

Summation of product of $m+1$ alternate numbers

We know that $$\begin{align} \sum_{r=1}^n \prod_{j=0}^m (r+j)&=\sum_{r=1}^n r(r+1)(r+2)\cdots(r+m)\\ &=(m+1)!\sum_{r=1}^n\binom {r+m}{m+1}\\ &=(m+1)!\binom {n+m+1}{m+2}\\ &=\frac ...
0
votes
1answer
33 views

Find the mean and variance of $V_n=\frac{1}{n}\sum_{i=1}^n(X_i-u)^2$

Suppose that $X_1,X_2,...,X_n$ is a random sample from a distribution with mean $\mu$ and variance $\sigma^2$. Suppose also that $v:=\mathbb{E}[(X_1-\mu)^4]<\infty$. Find the mean and variance of ...
0
votes
2answers
23 views

A question on summation notation and pi notation for multiplication.

As I am in high school, I know the basics to summation and pi notation. However when people put things other than numbers on the top and bottom of the summation, I do not understand what they mean. ...
1
vote
2answers
42 views

What is the correct mathematical notation for something comprised of the sum of constituents n where n is infinite?

I am trying to figure out what the correct mathematical notation would be for something like the following: I want to describe that the value V of a company is equal to sum of parameters P at any ...
2
votes
1answer
47 views

Help me approximate this sum: $S = \sum_{j=2}^{N}{\frac{\ln \ln \ln \ j}{( \ln \ln \ j)^2}}$

I would like to figure out the asymptotic rate of growth for the sum $S = \sum_{j=2}^{N}{\frac{\ln \ln \ln j}{( \ln \ln j)^2}}$ in the limit of large $N$. Ultimately, I want to know if $S(N)$ is ...
1
vote
1answer
38 views

Verifying the (real) sum of the series, $\sum_{n\in \mathbb{Z}}2^{-|n|}e^{inx}$ for all $x\in \mathbb{R}$.

I found the old solution of it and it says that $$\sum_{n\in \mathbb{Z}}2^{-|n|}e^{inx}=\frac{13-8\cos(x)}{5-4\cos(x)}.$$ I wondered it might be a mistake. Here's what I think the result should be ...
5
votes
0answers
44 views

Minimize $f(m)=\sum_{n=1}^\infty n^m / m^n $

For what real value of $m$ such that $\displaystyle\sum_{n=1}^\infty \frac{n^m}{m^n} $ is minimized? I've been told that it's equivalent of solving for $ \text{Li}_{-n}\left(\frac1n\right)$ for the ...
-4
votes
1answer
92 views

$\sum\limits_{n=1}^{\infty}\sin ( \frac{5^n + 2^n}{n!})$ converges? [closed]

I was trying to determine weather or not $\sum\limits_{n=1}^{\infty}\sin ( \frac{5^n + 2^n}{n!})$ converges using perhaps the D'Alembert test, but it doesn't really seem to fit..are there other ways? ...
3
votes
5answers
144 views

Simplifying sum equation. (Solving max integer encoded by n bits)

Probably a lack of understanding of basic algebra. I can't get my head around why this sum to N equation simplifies to this much simpler form. $$ \sum_{i=0}^{n-2} 2^{-i+n-2} + 2^i = 2^n - 2 $$ ...
-2
votes
1answer
58 views

$\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges? [closed]

I was trying to determine weather or not $\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges using perhaps the D'Alembert test, but given the sine I cant really see it happening..are there ...
0
votes
2answers
60 views

Identify $\sum\limits_{n=1}^{99 }\log (1 + \frac{1}{n})$

I'm looking for hints on how to compute $\sum\limits_{n=1}^{99}\log (1 + \frac{1}{n})$. The series in the log doesn't seem to be geometric or anything so there isn't any apparent way to sum it... ...
0
votes
3answers
75 views

$\sum\limits_{n=4}^{\infty } \frac{2^n + 8^n}{10^n} = ?$

im looking for hints on how to do: $\sum\limits_{n=4}^{n= \infty } \frac{2^n + 8^n}{10^n} = ?$ I thought this may have had something to do with geometric series but nothing obvious comes up ...
2
votes
1answer
26 views

How to represent the sum of matrix elements given all permutations of a set of indices?

I would like to represent the sum all matrix elements of all permutations of indices given a set. For example, given the set $S=\{1,2,3\}$ I would like to compactly express ...
0
votes
1answer
35 views

Summation of Infinite Areas of Triangles Involving Median

A triangle has an area of 2. The lengths of its medians equal the lengths of the sides of a second triangle. The lengths of the medians of the second triangle equal the lengths of the sides of a third ...
-2
votes
2answers
52 views

Is the sum of the following series a finite number or not? Explain. $ \sum_{k=1}^\infty \frac{5\sin^2k}{k!} $ [closed]

Is the sum of the following series a finite number or not? Explain. $$ \sum_{k=1}^\infty \frac{5\sin^2k}{k!} $$
2
votes
2answers
44 views

Determine the Value of $\sum_{n=0}^{\infty} (1+n)x^n$ [duplicate]

For $x\in\mathbb{R}$ with $|x|<1$. Find the value of $$\sum_{n=0}^{\infty} (1+n)x^n$$
3
votes
3answers
57 views

Bounding $\sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$

I'm looking for a bound depending on $N$ of $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$. The following holds $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}} = \sum_{k=1}^N ...
0
votes
1answer
55 views

Why does $ \sum_{i=0}^n \theta_i X_i = \theta^TX$ hold? [closed]

While reading Machine learning, I came through a formula, which is $$ \sum_{i=0}^n \theta_i X_i = \theta^TX .$$ I would like to know the name of the equation and some explanation behind it.
0
votes
3answers
44 views

summation for $x \gt 1$

how do I show that the $\sum_{n=0}^{n=N} (n+1)x^n$ is less than or equal to the square of this whole sum : $\sum_{n=0}^{n=N}x^n$ Tried induction didn't work. Got messy tried doing directly by using ...
1
vote
0answers
28 views

Simplifying Repeated Summation

Is it possible to simplify the expression below in order to reduce it to a single sum? Would Pascal's Triangle possibly be applicable to finding a solution? $$ \sum_{a_1=1}^n \sum_{a_2=1}^{a_1} ...
2
votes
3answers
98 views

Proving $\int_0^n \left(1-\frac{t}{n}\right)^n\ln(1/t)\,dt \to \gamma$

I have to prove that $\displaystyle \lim_{n\to\infty}\int_0^n \left(1-\frac{t}{n}\right)^n\ln(1/t)\,dt= \gamma$ I tried to expand $\left(1-\frac{t}{n}\right)^n$ and swap sum and integral, which ...
1
vote
5answers
98 views

How to calculate $\sum_{n=1}^\infty \frac{2}{(2n-1)(2n+1)}$ [closed]

How can I calculate the sum of the following infinite series: $$\sum_{n=1}^\infty \frac{2}{(2n-1)(2n+1)}$$
2
votes
0answers
13 views

A geometric-like sum with N variables

I want to compute the following sum $F_N=\sum_{1\le n_1<n_2\cdots<n_N\le L} a_1^{n_1} a_2^{n_2}\cdots a_N^{n_N}$ for rather large $L$. Is there a general formula for this kind of sum ?
0
votes
0answers
27 views

I attempt to combine Abel's summation with Hardy's inequality

Let $a(n)$ be a sequence of real numbers, $A(x)=\sum_{n\le x}a(n)$, with $A(x)=0$ if $x<1$ and $G(x)=\int_0^x g(t)dt$, with $g(t)\ge 0$ integrable on $[0, \infty)$, $p>1$ (is a requeriment for ...
0
votes
1answer
64 views

why doesn't proof of sum of two rational number is rational not proving the irreducibility of fraction $\frac{ad+bc}{bd}$?

When I was comparing proof for $\sqrt{2}$ and sum of two rational numbers, I found that the proof of two rational number did not mention anything about common factor in the ratio. one proof I found ...
3
votes
7answers
112 views

How do you solve the summation of $2-4+8-16+32- \dots 2^{48}$?

This is a summation problem but I can't seem to figure out how to solve this with the mix of subtraction and addition.
0
votes
0answers
31 views

How to calculate sum of sum of subsets using polynomial multiplication?

Recently I was reading about polynomial multiplication, and came across solution of one interesting problem. Which is finding sum of product of all $k-subsets$. which is as following: if we have ...
0
votes
1answer
33 views

Linearity of the supremum

Short question: In which cases we have: Let $f_n$ be a sequence of functions, then $\sup_{k\in \mathbb N}(\sum_{n}^if_k(n))=\sum_{n}^i\sup_{k\in \mathbb N}f_k(n)$ ? I guess if the $f_k(n)$ are ...
0
votes
5answers
89 views

$\sum_{i=0}^n {2n \choose 2i} = 2^{2n-1}$

$$ \sum_{i=0}^n {2n \choose 2i} = 2^{2n-1} $$ I know what this sum is supposed to equal. I also have a hint that I am supposed to use ${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r}$ I was ...
0
votes
0answers
31 views

How can I solve for X in a sum function?

I have this inequality constrain, let's call it (*) $$\frac{1}{C}\sum_{i=1}^{n} X_{i}Y_{i} < Z$$ (forgive me if it should be $i=0$ and $n-1$, this is my first question here and I'm very rusty) ...
2
votes
5answers
100 views

How to sum:$\sum_{k=1}^{n}(k²-k)$ without using this: $\sum_{k=1}^{n}(k²)=\frac{n(n+1)(2n+1)}{6}$?

I have tried to find the sum of this series $\sum_{k=1}^{n}(k^2-k)$ without using $\sum_{k=1}^{n}(k^2)=\frac{n(n+1)(2n+1)}{6}$ but i can't . Can anyone explain this to me if there is a way to do ...
2
votes
2answers
50 views

Simplify sum $\sum_{i=0}^k(-1)^ii\binom{n}{i}\binom{n}{k-i}$ for $n\geq k\geq 0$

The problem asks us to simplify the following sum: $$\sum_{i=0}^k(-1)^ii\binom{n}{i}\binom{n}{k-i}$$ for $n\geq k\geq 0$. I've tried the following: ...
7
votes
1answer
52 views

Partitioning $n$ naturals summing $2N$ into two sets summing $N$

I'm trying to solve this problem: Let $a_1, \ldots , a_n$ be natural numbers such that $a_k \le k$ for every $k = 1,\ldots,n$, and $\sum_{k=1}^{n} a_k=2N$. Show that there exists a partition of ...
3
votes
1answer
49 views

Find this closed form $\sum_{k=1}^{n}\left(\lfloor a_{k}\rfloor +\lfloor a_{k}+\frac{1}{2}\rfloor \right)$

Let $\dfrac{1}{a_{k}}=\dfrac{1}{k^2}+\dfrac{1}{k^2+1}+\cdots+\dfrac{1}{(k+1)^2-1}$ I need some ideas to exploit for finding the closed form of $$\sum_{k=1}^{n}\left(\lfloor a_{k}\rfloor +\lfloor ...
2
votes
0answers
38 views

Summation Formula Interpretation involving Roots of Unity

In a paper on Lacunary Recurrence Relations, D. H. Lehmer is beginning his preliminary information and presents the following sum formula: Let $p,q,r,s$ be positive integers, and let $n,t$ be ...
2
votes
1answer
73 views

Sum with binomial coefficients and integer powers

I would like to have an analytic expression for the following sum $$ G_{n,a} = \sum_{p=1}^n \frac{(-1)^p p^{2(a+n)}}{(n-p)! (n+p)!} \;. $$ I am not sure it has a closed form, but I would at least ...
1
vote
1answer
19 views

if $Y\sim \textrm{Bin}(n,\alpha) \space,\space X\mid Y\sim \textrm{Bin}(Y,\beta)$ then $X\sim \textrm{Bin}(n,\alpha \beta)$

Suppose \begin{equation*}Y\sim\textrm{Bin}(n,\alpha)\space,\space (X\mid Y)\sim\textrm{Bin}(Y,\beta)\end{equation*}I'm trying to prove that $X\sim\textrm{Bin}(n,\alpha \beta)$. I began with saying ...
3
votes
3answers
51 views

Computing sum with $\cot$

I want to compute terrible sum: $$ z\cot z = \frac{z}{2^n}\left(\cot\frac{z}{2^n}-\tan\frac{z}{2^n} + \sum_{k=1}^{2^{n-1}-1} \cot\left(\frac{z + k\pi}{2^n}\right) + \cot\left(\frac{z - ...
7
votes
3answers
68 views

Closed form for the sum $\sum_{k=0}^n a^k \left\lfloor\frac{k}{p}\right\rfloor$

I want a closed form for the sum $$S=\sum_{k=0}^n a^k \left\lfloor\frac{k}{p}\right\rfloor$$ where: $a\ne 1<p<n;\quad p\in\mathbb Z$ I know a related identity, $$\quad\displaystyle ...
5
votes
4answers
48 views

How to determine the number removed from the list [duplicate]

One number is removed from a set of integers from 1 to n,the average of the remaining numbers is $\large{\frac{163}{4}}$. Which number was removed? I tried to find the mean of ...
0
votes
2answers
82 views

Prove by induction that $\sum_{i=1}^{n} 2i=(n+1)n$, for every positive integer n. [duplicate]

Can anyone explain the concept behind this? I just don't get how I should proceed with it? Like each step, why and how is it done? Prove by induction that $\displaystyle\sum_{i=1}^{n} 2i=(n+1)n$, ...