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

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2
votes
1answer
54 views

Summation with Zeta function

I'm currently studiyng Zeta function and I don't understand this identity: $$\sum_{n=1}^\infty x \sum_{p=0}^\infty \frac{x^{2p}}{(n\pi)^{2p+2}} = \sum_{p=1}^\infty \pi^{-2p}\zeta(2p)x^{2p-1} $$ I ...
3
votes
1answer
46 views

How does $\sum_{k=0}^n (pe^t)^k{n\choose k}(1-p)^{n-k} = (pe^t+1-p)^n$?

How does $\sum_{k=0}^n (pe^t)^k{n\choose k}(1-p)^{n-k} = (pe^t+1-p)^n$? Where $e$ is Euler's number and $p,n$ are constants. ${n\choose k}$ is the binomial coefficient. If context helps, I'm ...
10
votes
1answer
438 views

New Year Combinatorics

In the spirit of the festive period and in appreciation of the encouraging response to my X'mas Combinatorics problem posted recently, here's one for the New Year! Express the following as a ...
0
votes
2answers
46 views

Arithmethic Progression problem!

Given that the first term of an arithmetic progression is $5$ and the sum of the first eight terms is equal to the sum of the following four terms, find the common difference. What do they mean by " ...
1
vote
3answers
40 views

How to evaluate a sum which contains limit variables?

For example: $$\lim_{n\to\infty}\sum_{i=1}^n\frac{n-1}n\frac{1+i(n-1)}n $$ And would the result necessarily be rational, because each term appears to be the multiplication of two rational fractions? ...
1
vote
3answers
101 views

Summation of series with terms $U_n=\frac{1}{n^2-n+1} -\frac{1}{n^2+n+1}$

Given that $U_n=\dfrac{1}{n^2-n+1} -\dfrac{1}{n^2+n+1}$, find $S_N$= $\sum_{n=N+1}^{2N}U_n$ in terms of $N$. Find a number $M$ such that $S_n<10^{-20}$ for all $N>M$. I was able to ...
3
votes
1answer
58 views

Sum of squares of Binom(n,p) values

Let $x_{n,p}(j)$ be the probability that a random variable distributed according to a binomial distribution with parameters $n \in \mathbf{N}_+$ and $p \in (0,1)$ takes the value $j \in ...
-1
votes
1answer
99 views

Rational summation of irrational numbers

Is the sum of all irrational numbers between any two integer constants rational? I think it should be, because every irrational number should have another irrational with which it would sum to a ...
3
votes
1answer
123 views

Combinatoric Coefficients of a Polynomial

I have the following function: $$f(x)=\left(T_{N_2}(x)-T_{N_1}(x)\right)\left(T_{N_3}(x)-T_{N_1}(x)\right)\left(T_{N_3}(x)-T_{N_2}(x)\right)$$ where $T_{N}(x)=1+x+\frac{x^2}{2!}+...+\frac{x^N}{N!}$ ...
2
votes
0answers
31 views

Sum of first $n$ integer values of a polynomial as a polynomial in $n$

Let $p$ be a polynomial. Let $$S_p(n)=\sum_{k=1}^n p(k)$$ be the sum of the polynomial's values at the first $n$ integers. Splitting the sum along each term's monomials, pulling the common ...
4
votes
1answer
76 views

Summation of a multiple series involving Fibonacci numbers

Compute the sum $$\sum_{a_{2015} = 0}^{\infty} \sum_{a_{2014} = 0}^{a_{2015}} \sum_{a_{2013} = 0}^{a_{2014}} \cdots \sum_{a_{1} = 0}^{a_2} \sum_{k=0}^{a_1} \frac{F_{k}}{2^{a_{2015}}} $$ where $F_k$ ...
2
votes
2answers
95 views

Summation of the telescoping series $\sum_{n=1}^N \frac{x}{(1+(n-1)x)(1+nx)}$

$(i)$ Verify that $$\frac{1}{1+(n-1)x} - \frac{1}{1+nx} = \frac{x}{(1+(n-1)x)(1+nx)}$$ $(ii)$ Hence show that for $x \ne 0$, $$\sum_{n=1}^N \frac{x}{(1+(n-1)x)(1+nx)}=\frac{N}{1+Nx}$$ Deduce that the ...
1
vote
4answers
62 views

Sum of serie of integer that are halving

I'm trying to calculate the sum of integers that are halving (when the number is odd, we round down). Here are some example: S(100) = 100 + 50 + 25 + 12 + 6 + 3 + 1 + 0 + 0 + 0 + 0 + 0... S(3) = 3 ...
5
votes
0answers
75 views

How to simplify $\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$

Let $$x=\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$$ where $n,k\in\mathbb{Z}^+$. How to simplify $x$? I simplified it for $k=1,2,3$ and I got $n$, $\dfrac12n(n+1)$ ...
0
votes
1answer
60 views

How do we justify splitting a sum over $1 \le j < k+j \le n$ into two sums $1 \le k \le n$ and $1 \le j \le n-k$?

Can anyone justify the following summation manipulation? $$\sum_{1 \le j < k+j \le n} \frac{1}{k} = \sum_{1 \le k \le n}\,\sum_{1 \le j \le n-k} \frac{1}{k}$$
3
votes
3answers
81 views

How to prove $\sum_{k=1}^{n}k\binom{n}{k} = n2^{n-1}$ [duplicate]

$\sum_{k=1}^{n}k\binom{n}{k} = n2^{n-1}$ I have tried both induction and transforming both sides to get equality but no luck I know that $\sum_{k=1}^{n}\binom{n}{k} = 2^{n}-1$ and ...
0
votes
1answer
32 views

A Fourier series' upper bound involving gamma function

I am reading Donald E. Knuth's "The Art of Computer Programming" Vol. 3 and stuck on the equation 47 and inequality 48 on Page 133,which are the follows: $$ \delta(n)=\frac{2}{\ln2}\sum_{k \ge 1} ...
0
votes
1answer
52 views

Concrete Mathematics: How do we figure out the constrains of summations when using multiplication by summation factor method?

In chapter 2.2 of Concrete Mathematics, the authors introduced the usage of summation factor to convert recurrence to summation. The idea is to multiply $s_n$ on both sides of the recurrence relation ...
5
votes
2answers
77 views

How find this sum $\sum_{k=0}^{n}\binom{n}{k}|n-2k|$ closed form or asymptotic behaviour?

Find the following series closed form or asymptotic behaviour $$\dfrac{\displaystyle \sum_{k=0}^{n}\binom{n}{k}|n-2k|}{2^n}$$ I use wolfram can't give the closed form: see wolfram ,so I think ...
0
votes
1answer
92 views

Changing the order summation and limit and proving a double-sequence identity

As a part of a work of mine I wanna use this claim (which I hope is true), and don't know why I can: Assume I have for every $i\in \mathbb N$ a series $\{a_i^n\}_{n\in\mathbb N}\subset\mathbb R$ ...
1
vote
0answers
32 views

Cauchy Product with reindexing

Suppose I have the following: $$xe^{xz}A(x,z)$$ where $$A(x,z)=\sum_{n=0}^{\infty}\frac{a_n(z)}{n!}x^n$$ and $a_n$ is a polynomial in $z$ Taking the Cauchy product (since ...
1
vote
2answers
85 views

Geometric Series Word Problem from Khan Academy

A Sierpinski triangle, starts with a white equilateral triangle with sides of length $1$. First, the middle triangle is colored green. At the second step, 3 triangles are colored blue. At the third ...
0
votes
1answer
43 views

Proving that $\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}]$

I'm trying to prove that $$\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}\left[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}\right]$$ for $p,n \geq 2$, $p, q \in \mathbb{N}$. I'm trying to use induction ...
0
votes
3answers
35 views

Sum of series by comparing to a known Series expansion

I can't seem to quite grasp Summation of Series any bit and would like to ask for your help. The problem requires me to find the sum of series by using known series expansions. I have given the ...
2
votes
2answers
74 views

An identity that comes from computing the Wiener index of a cyclic graph

Can the below identity be proven in such a way that we can generalize it? $(1 + 1 + 2 + 2 + 3 + 3 + 4) +( 1 + 2 + 2 + 3 + 3 + 4) + (1 + 2 + 3 + 3 + 4)+ +( 1 + 2 + 3 + 4 )+(1 + 2 + 3) + (1 + 2) + 1 = ...
2
votes
1answer
35 views

Prove $\sigma(\tau(I))=(\sigma\tau)(I)$

$I=(i_1,...,i_k)$ denotes an ordered $k$-tuple of indices. Given $\sigma\in S_k$, define $\sigma(I)=(i_{\sigma^{-1}(1)},...,i_{\sigma^{-1}(k)})$ Let $\sigma,\tau\in S_k$. Then ...
0
votes
1answer
27 views

Show $P(\sigma\tau)=P(\sigma)P(\tau)$

$\forall\sigma,\tau\in S_n$, $P(\sigma\tau)=P(\sigma)P(\tau)$ Definition: To each $\sigma\in S_n$ we may associate an $n\times n$ permutation matrix $P(\sigma)$ given by ...
27
votes
1answer
1k views

X'mas Combinatorics

Inspired the various** algebraic X'mas greetings sent to me over the festive period, I thought I would try to devise one of my own. $$\Large ...
1
vote
1answer
60 views

Strange Sigma Notation

How do I interpret this form of sigma notation? Do e1 and e2 take on all combinations of 1 and -1? If they do, what's the point? They just get multiplied inside the sum! FYI, this comes from equation ...
6
votes
6answers
171 views

How to compute $\sum^n_{k=0}(-1)^k\binom{n}{k}k^n$

When trying to answer this question I arrived at $$\int^\infty_0\frac{\sin(nx)\sin^n{x}}{x^{n+1}}dx=\frac{\pi}{2}\frac{(-1)^n}{n!}\sum^n_{k=0}(-1)^k\binom{n}{k}k^n$$ After using Wolfram Alpha to ...
6
votes
1answer
66 views

An identity involving partial fractions decompositions

In Vladimir A. Smirnov's book Analytic Tools for Feynman Integrals (page 38), the following identity is suggested to perform partial fractions decompositions $$ \begin{split} ...
3
votes
0answers
69 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 ...
1
vote
1answer
35 views

The alternating sum of primes defines an injection

Define $\displaystyle\alpha(n)=\sum^n_{k=1}(-1)^{n-k}p_k$, where $p_k$ is the $k$:th prime. Show that $\alpha$ is an injection $\mathbb Z_+\to\mathbb Z_+$. It's easy to see while considering sums as ...
0
votes
0answers
40 views

Show this equality is equal [duplicate]

![show this equality is equal][1] the equality is $$\frac{(1+2x+x^2) (1+x)}{(1-2x+x^2)(1-x)} = 1 + [ 6x + 18x^2 + 38x^3 + 66x^4 +\ldots ]$$
0
votes
3answers
92 views

find sum of ln(1- $\frac{1}{n^2})$. prove it converges [duplicate]

![enter image description here][1] $$ \sum = \ln(1 - \frac{1}{4} ) + \ln(1 - \frac{1}{9} ) + \ln ( 1 - \frac{1}{16}) = \ln(\frac{3}{4}) + \ln (\frac{8}{9}) + \ln ( \frac{15}{16}) = -.287 + -.117 + ...
0
votes
4answers
144 views

How to prove that the sum of squared binomials equals $\binom{2n}{n}$ [duplicate]

I've stumbled upon this lemma a few times in my textbook: $$\sum_{k=0}^{n}\begin{pmatrix}n\\k\end{pmatrix}^2=\begin{pmatrix}2n\\n\end{pmatrix}$$ I've been trying to prove it, but I simply can't seem ...
2
votes
2answers
17 views

How does $F_j \frac{\partial F_j}{\partial x_i} = \frac{1}{2} \frac{\partial(F_j F_j)}{dx_i}$

How does $$F_j \frac{\partial F_j}{\partial x_i} = \frac{1}{2} \frac{\partial(F_j F_j)}{dx_i}$$ For $\mathbf{F}(x_1,x_2,x_3)$ being a continuously differentiable vector field.
1
vote
1answer
27 views

Show that $\nabla \times (\mathbf{c}\times \mathbf{r}) = 2\mathbf{c}$

Suppose $r=x_i\mathbf{e}_i$ I need to do this question using the Einstein summation convention. So far I have: $\nabla \times (\mathbf{c}\times \mathbf{r}) = \varepsilon_{ijk} ...
0
votes
0answers
44 views

How to add a sum of sines with general form $\frac{(4n-2)\pi}{23}$ between $1$ and $11$?

The question is to solve for $x$: $$\tan x = \sum_{n=1}^{11} \sin\frac{(4n-2)\pi}{23}$$ How do I express the above as a tangent of one angle without using a calculator In general what is the ...
0
votes
0answers
73 views

Nice approximations of sums by integrals.

Let $f(x):\Bbb Z^+\rightarrow \Bbb R^+$ be a non-monotone function. If for every $m\in\Bbb N$, $$S(m) =\sum_{n=1}^N\frac{1}{(1+f(n))^m}$$ be sum of interest, then is there a way to approximate this ...
1
vote
2answers
62 views

Summation of series

If the sum $$\sum_{n=0}^{2011} \frac{n+2}{n!+(n+1)!+(n+2)!}$$ can be written as $$\frac{1}{2} - \frac{1}{a!}$$find the last three digits of a. I have reduced the given expression to ...
1
vote
3answers
71 views

Why does $1+p+p^2+\dotsb+p^{n-1}=\frac{1-p^n}{1-p}$ [duplicate]

$$y_n=\rho^ny_0+(1+\rho+\rho^2+\cdots+\rho^{n-1})b.$$ If $\rho \not=1$, we can write this solution in the more compact form $$y_n=\rho^ny_0+\frac{1-\rho^n}{1-\rho}b.$$ This is from Elem. Diff. ...
-1
votes
4answers
53 views

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$?

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$? I know $\lim_{n \to \infty} \sum_{k=0}^n\frac{t^k}{k!}=e^t$, but my sum starts at $k=1$ and also has ...
1
vote
0answers
41 views

Given a closed form for a series, what can be said about the sum of the squares of its terms?

Suppose I have an infinite integer sequence $\{a_k\}$, and suppose I know a closed form in terms of $n$ for this sum: $$\displaystyle\sum\limits_{k=1}^{n} a_k$$ Given this, is it always (or ever) ...
3
votes
1answer
83 views

formula for summation notation involving variable powers

I need help finding the formula for this summation notation: $$\sum_{k=1}^n{k^{2k} }$$ or $$1^2 + 2^4 +3^6 +.....+n^{2n} $$ And preferably not involving calculus.
2
votes
2answers
48 views

If given $\sum_{r=1}^{m-1}\binom r3$, how does the summation evaluate when $n<r$ in $\binom nr$?

Correct me if I'm running the summation correctly - $$\sum_{r=1}^{m-1}\binom r3=\binom 13+\sum_{r=2}^{m-1}\binom r3$$ $$\sum_{r=1}^{m-1}\binom r3=\binom 13+\binom 23+\sum_{r=3}^{m-1}\binom r3$$ ...
0
votes
4answers
52 views

Why is $\sum_{r=1}^{m-1} (2r+1)r=\sum_{r=1}^{m-1} 4\binom{r}{2} + 3\binom{r}{1}$?

How did the summation expression get transformed to combination? From where did the constants $4$ and $3$ come from? $$ \begin{align*} T(m^2-1) &= \sum_{r=1}^{m-1} (2r+1)r\\ &= \ ...
1
vote
2answers
44 views

What property of summation is used while solving this problem?

Saw this problem on a website. Can someone explain how the summation is split into summation of summation? What property of summation was used here? $$ T(n) = \sum_{k=1}^n \lfloor \sqrt{k} \rfloor. $$ ...
0
votes
1answer
44 views

Quicksort-How did we get the relation?

At the proof of the theorem that the expected time of Quicksort is $O(n \log n)$, there is the following sentence: We suppose that the partitions are equally likely, so the possibility that the sizes ...
1
vote
1answer
39 views

Find a probability of $n$ event happening from $m$ types

The question is: to find a sum $$ S=\sum\limits_{n_1+n_2+\ldots+n_m = n,\ n_i=0,1,\ldots,n} p_1^{n_1}p_2^{n_2}\cdots p_m^{n_m}, $$ where $p_i\in[0,1]$. UPDATE. This issue has no probabalistic ...