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

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0
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1answer
36 views

Inequality in non-decreasing sequence

Let $a, b$ be two sequences of real numbers such that $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$. Prove (or disprove) that ...
3
votes
1answer
50 views

Cauchy-Schwarz inequality on double-summation term

I have the following, where $v$ is a vector $$ v\cdot (v\cdot \nabla)v $$ which in index notation becomes $v_jv_id_iv_j$. I want to apply the Cauchy-Schwarz inequality on this, which is given by $$ ...
0
votes
3answers
78 views

partial Fibonacci summation

Let $F_{n}$ be the n-th Fibonacci number. How to calculate the summation like following: $\sum_{n \geq 0} F_{3n} \cdot 2^{-3n}$
1
vote
1answer
62 views

Least common multiple in binomial expansion

If I sum the terms of a binomial expansion, which would be the least common multiple of all the denominators? Say $\displaystyle \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \ldots + \binom{n}{n}$ ...
2
votes
2answers
83 views

Totient function sum over divisors

I would like to know if there is a closed form solution for $$G(n)=\sum\limits_{d\mid n}(-1)^{\frac{n}{d}}\phi(d)$$ It seems quite likely there is since $$\sum\limits_{d\mid n}\phi(d)=n$$ But I ...
0
votes
1answer
42 views

How prove this $arg((1+ia)(2+ia)(3+ia)\cdots(n+ia))=\arctan{\frac{a}{1}}+\arctan{\frac{a}{2}}+\cdots+\arctan{\frac{a}{n}}$

let $i^2=-1,a>0$, show that $$arg((1+ia)(2+ia)(3+ia)\cdots(n+ia))=\arctan{\dfrac{a}{1}}+\arctan{\dfrac{a}{2}}+\cdots+\arctan{\dfrac{a}{n}}$$ I can't How prove this equation,Thank you because ...
2
votes
1answer
50 views

Is this summation $> 0$ or $< 0$?

Sum: $$ s =\frac{\left(\sum_{i=1}^n a_i^{p-1} v_i\right)\left(\sum_{i=1}^{n} a_i^{p-1} v_i\right)}{\sum_{i=1}^{n} a_i^p} - \sum_{i=1}^{n} a_i^{p-2} v_i^2 $$ Where $0<p<1$ and $a_i, v_i$ are ...
0
votes
0answers
51 views

Sum of positive numbers yields negative answer? [duplicate]

In my algorithms class, my professor just briefly informed us that $$\sum\limits_{i=1}^\infty i= -\frac{1}{12}$$ might hold true. Every single thing I have learned about mathematics up until this ...
3
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0answers
54 views

Find generating function For sequences

Can anyone out here help? The exercise says: "Find the generating function for each of the sequences below (the general term is given)" Now, the question is how do you find one for those: a) $U_n = ...
0
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0answers
26 views

Proof sum of permutation

I'm trying to prove: $$P(N) = \sum permutation(A,N)=1 \tag{1}$$ for the particular choice of the set $A = \{ \mu_1, \dots, \mu_n, 1-\mu_1, \dots, 1-\mu_n \}$, where $i = 1, \dots, N$ . So for ...
3
votes
2answers
78 views

Sum of finite series of $\exp(an^2 + bn + c))$

Is there a way to simplify that sum to an expression without actual performing the summation, similar to the formula for calculating the sum of a (finite) geometric series? $\sum_{n=0}^{N-1} ...
1
vote
1answer
54 views

How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$?

How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$? This counts the number of subsets from $\{1,2,3,\dots,n\}$ having size $7$. To me, the summation part counts subsets of size ...
1
vote
1answer
19 views

How to interpret this summation equation

I'm a computational biologist trying to interpret this equation correctly. $$CI(x)= 1- \frac{2}{(N(N-1))} \sum_i^N\sum_j^N [ S(x_i,x_j) / \sqrt{S(x_i,x_i)S(x_j,x_j)}\ ] $$ i=1 ; j=i+1 I'm confused ...
1
vote
2answers
30 views

Randomized Algorithm

I asked this question earlier but I wanted to change the problem. A band has tour sites A, B, and C. They get paid every time they play at each tour site, specifically: ...
1
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2answers
52 views

Is $\sum_{x=1}^n (3x^2+x+1) = n^3+2n^2+3n$?

I wanna check if the following equation involving a sum is true or false? How do I solve this? Please help me. $$ \sum_{x=1}^n (3x^2+x+1) = n^3+2n^2+3n$$ for all $n \in \{0,1,2,3, \dots\}$.
1
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2answers
34 views

Finding rooms of Summation

Hi I was wondering how do I Solve this question. I have to solve for $a$. I can solve for it when there's one summation but now there are three. My guess is factoring out the $A$. Divide $s$ by the ...
3
votes
1answer
149 views

Sum this series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\ldots$ upto $n$ terms

Sum this series: $$\dfrac{1}{1+1^2+1^4}+\dfrac{2}{1+2^2+2^4}+\ldots$$ upto $n$ terms. My approach: $$(1-n^6)=(1-n^2)(1+n^2+n^4)\implies \dfrac{n}{1+n^2+n^4}=\dfrac{n(1-n^2)}{1-n^6}$$ So, the ...
0
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1answer
34 views

How to prove that convolution on real sequences is associative?

Given two real sequences $\{ a_n \}$ and $\{ b_n \}$, where $n \ge 0$, the convolution operation (denoted $\ast$) is defined as $\{ c_n \} = \{ a_n \} \ast \{ b_n \}$, where $c_n = \sum_{k=0}^{n} a_k ...
2
votes
1answer
51 views

Help Obtaining an Asymptotic for $\sum_{n=1}^{m-1}\ln(n)\ln(m-n)$

How can I obtain an asymptotic for this partial sum, with an error term of at most $O(m^{1/2}\ln(m))$: $$\sum_{n=1}^{m-1}\ln(n)\ln(m-n)$$ I tried flipping the order of summation half way through to ...
1
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3answers
76 views

How do I compute this triple summation?

$$\sum_{i=0}^{n-1} \sum_{j=0}^{i-1} \sum_{k=0}^{j-1} i + j + k$$ The question is looking for a $\Theta(g(n))$ function to represent this summation, but I am uncertain how to go about computing triple ...
-1
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1answer
30 views

Question On Summation [closed]

This Summation question is a bit advance for me, I was wondering if anyone could help me derive the solution for this,
1
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2answers
55 views

Double summation, what is the right function?

I want to write a summation to count the total number of number of books sold from a list of B different books by P different publishers. That is, there is one list of books and each publisher can ...
0
votes
1answer
35 views

What is the series expansion of $f(z)\cdot\exp\left({s\,\log(z)}\right)$?

For analytic $f$, how can I represent the expression $f(z)\cdot\exp\left({s\,\log(z)}\right)$, i.e. $f(z)\cdot z^s$ in the form $$\sum_{n}^\infty\left(\sum_{k}^\infty a_k s^k\right)z^n,$$ at least ...
4
votes
1answer
45 views

How to prove this formula using sums?

I need to prove this formula $$\sin x\cos y+\cos x\sin y=\sin(x+y)$$ using sums $$\sin x=\sum_{n=0}^{\infty }\frac{(-1)^n\cdot x^{2n+1}}{(2n+1)!}$$ $$\cos x=\sum_{n=0}^{\infty }\frac{(-1)^n\cdot ...
1
vote
1answer
56 views

How do I express the integral of $\tan^{2n}(x)$ using sigma notation?

How do I express the integral of $\tan^{2n}(x)$ with respect to $x$ using sigma notation? $$\int \tan^{2n}(x)\, dx = \text{???}$$
2
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0answers
55 views

Find the sum of exponentails of squares $\sum_{r=1}^n e^{-\alpha r^2}$

I would like to find $$a_n =\sum_{r=1}^n e^{-\alpha r^2},\qquad \alpha\in\mathbb{R}$$ I tried to solve the equivalent recursion $$a_n=a_{n-1}+e^{-\alpha n^2}\quad(n>0),\qquad a_0=0.$$ with an ...
7
votes
4answers
301 views

Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\ldots+\frac{1}{3001}<\frac43$

Prove that $$1<\dfrac{1}{1001}+\dfrac{1}{1002}+\dfrac{1}{1003}+\ldots+\dfrac{1}{3001}<\dfrac43 \, .$$ My work: $$\begin{eqnarray*} ...
0
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0answers
25 views

About complex exponential summation

Let $f:\,\mathbb{R}^{+}\rightarrow\mathbb{R},\, f\in C^{\infty}\left(\mathbb{R}^{+}\right)$ and such that $f\left(n\right)>0\,\forall n\in\mathbb{N}$. Let $c>0$ a real number, $N>0$ a large ...
0
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1answer
66 views

Digit in units place of 1!+2!+…99!

There isn't much I can add to the question description to expand upon the title. I came across this in a multiple choice test. The options were 3, 0, 1 and 7. I am absolutely stumped. Any pointers? By ...
0
votes
2answers
56 views

finite sum over a Gaussian

I have a sum of the form: $$\sum_{n,m=-N}^N e^{-\alpha (n-m)^2}$$ where $\alpha > 0$ is some constant, and $n,m$ take the integer values: $-N,..,N$. I know there is a possibility of exchanging ...
1
vote
1answer
25 views

Summation notation for time series

I need to add together the results of a function for four consecutive years prior to the start of a project. The function is calculated for individual years. It is (10 × (A + B + C) ÷ ...
2
votes
2answers
31 views

About complex sum and modulus

Let $\left(a_{n}\right)_{n},\,\left(b_{n}\right)_{n}$ two succession of non negative real numbers, $\left(c_{n}\right)_{n}$ a succession of complex numbers and $N$ a large natural number. Suppose that ...
0
votes
3answers
59 views

Using Cauchy-Schwarz inequality to prove that the mean of n real numbers is less than or equal to the root-mean-square of those numbers

Expressed mathematically, the question is to prove the that $\frac{1}{n}$ $\sum_{i=1}^{i=n}{a_i}\leqslant$ $\sqrt{\frac{1}{n}\sum_{i=1}^n{x_i}^2}.$ First of all, what form of Cauchy-Schwarz should I ...
-11
votes
1answer
118 views

1+2+3+4+… = -1/12 [duplicate]

I was browsing through Youtube and I saw this really cool thing.. but it seems really counter-intuitive could someone please explain to me why: 1+2+3+4+..... = -1/12? Here's the link to the guy's ...
5
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2answers
269 views

Proving $\sum_{k=1}^n{2k-1\choose k}{2n-2k+1\choose n-k+1}=4^n-{2n+1\choose n+1}$

Some background. I was asked to find an arithmetic function $f$ such that $f*f=\mathbf 1$ where $\mathbf 1$ is the constant function 1 and $*$ denotes Dirichlet convolution. I was able to prove that ...
3
votes
2answers
56 views

Showing that $\sum \sqrt{a_na_{n + 1}}$ converges given that $\sum a_n$ converges [closed]

Suppose a series $\sum a_n$ of nonnegative reals converges; show that $\sum \sqrt{a_na_{n + 1}}$ also converges.
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3answers
75 views

Explanation of sin(x) and cos(x) [closed]

Can anyone explain me what is this equation telling us? I need to implement it in my computer program :P AND
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4answers
309 views

Tricky summation question

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate $$ \sum^{n}_{i\mathop{=}0}\frac{1}{n+k+i}\cdot\frac{(m+n+i)!}{i!(n-i)!(m+i)!}$$ Any hints? I'm stuck on ...
5
votes
1answer
79 views

Hockey Classics at Matheletics '13

I'm trying to solve a challenge from Matheletics '13: Micheal Nobbs is organizing a training camp for identifying new talents in Indian Hockey. The camp witnessed a total of ($3K+1$) players. Each of ...
1
vote
1answer
53 views

If $tr(A+B)>tr(A)$, does it hold that $tr((A+B)^k)>tr(A^k)$ for all $k\geq 1$

I wonder whether the following holds and if so how it could be proved: Let $A, B$ be (non-commuting) positive semi-definite matrices, If $tr(A+B)>tr(A)$, does it hold that ...
1
vote
1answer
73 views

Simplification of a nested sum

I have a nested sum like so: $$\underbrace{\sum_{k_1=k_0}^{k^*} \ ... \sum_{k_n=k_{n-1}}^{k^*}}_{\text{n times}} 1\quad\ \text{with}\ \ n, k_0, k^* \in \mathbb{N},\ k^*\geq k_0$$ Is there a general, ...
0
votes
1answer
71 views

Can this summation be simplified?

I got something like $$ a_{n} = {1 \over 4^{n + 1}}\sum_{k = 0}^{\left\lfloor n/2\right\rfloor} {n + 1 \choose 2k + 1}\left(-3\right)^{k} $$ Could this be simplified more?
0
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1answer
30 views

Notation for nested sigmas (summations)

Is there any standard notation, other than an ellipsis, for a chain of nested sigma summations? For instance, I have: $$ \sum_{b_0=0}^{L} \sum_{b_1=0}^{L-b_0} \sum_{b_2=0}^{L-b_0-b_1} \cdots ...
4
votes
2answers
50 views

Simplification of Nested Summations

Given the following code: for i = 1 to n for j = 1 to i for k = j to (i+j) r = r + 1 end end end print r I get: $$ ...
1
vote
2answers
115 views

Summation of factorials modulo ten

I have read that$$\sum\limits_{i=1}^n i!\equiv3\;(\text{mod }10),\quad n> 3.$$ Why is the sum constant, and why is it $3$?
2
votes
0answers
76 views

How to prove this combinatorial identity

I am wondering how to prove the following identity: $$\sum_{k=0}^r {r-k \choose m} {s \choose k-t} (-1)^{k-t} = {r-t-s \choose r-t-m}$$ It seems that I can negating the upper index of ${s \choose k-t} ...
1
vote
4answers
105 views

Show $1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac{1}{2}+\frac{\sin[(n+1/2)θ]}{2\sin(θ/2)}$ [duplicate]

Show $$1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac12+\frac{\sin\left(\left(n+\frac12\right)θ\right)}{2\sin\left(\frac\theta2\right)}$$ I want to use De Moivre's formula and ...
18
votes
4answers
287 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
0
votes
0answers
37 views

Determining the disc of convergence in two series and determining at which points on the boundary of the disc the series converges.

The two series are as follows: $f(z) = \sum\limits_{n = 1}^\infty n(z+1-i)^{2n}$ and $f(z) = \sum\limits_{n = 1}^\infty n^{-1}z^{n}$ I have worked out that the discs of convergence are, ...
0
votes
3answers
128 views

Sum of $k {n \choose k}$ is $n2^{n-1}$

Proof that $\suṃ̣_{k=1}^{n}k {n \choose k}$ for $n \in \mathbb N$ is equal to $n2^{n-1}$. As a hint I got that $k {n \choose k} = n {n-1\choose k-1} $. I tried solving this by induction but, in the ...