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

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18 views

Summation of gcd of pairs

I want to find the mathematical formula for this pseudo-code ans = 0 for(i = 1 to x) { for(j = 1 to y) ans+= gcd(i, j) } print ans Please help me ...
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3answers
62 views

Prove that : $\sqrt C_1+\sqrt C_2 +\sqrt C_3 \ … +\sqrt C_n \leq \sqrt{n(2^n-1)}$

If $ C_0, C_1 , C_2, ... , C_n$ are the combinatorial coefficients in the expansion of $(1 +x)^n$, $n\in N$, then prove the following :$$\sqrt C_1+\sqrt C_2 +\sqrt C_3 \ ... +\sqrt C_n \leq ...
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0answers
52 views

Identity of two sums

I was trying to solve https://erdos.sdslabs.co/problems/31 The positive real numbers $x_0,x_1,x_2,\dotsc,x_m$ satisfy $x_0=x_m$ and $x_{i-1}+\frac{k}{x_{i-1}} = kx_i+\frac{1}{x_i}.$ Let ...
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3answers
108 views

A sum related to binomial theorem

If $\dfrac{x^2+x+1}{1-x} = a_0+a_1x+a_2x^2+\cdots$ then $\displaystyle\sum_{\gamma = 1}^{50}a_{\gamma} = ??$ Original Image This is a sum related to evaluating a series, from the chapter ...
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1answer
45 views

Expected score of a game [duplicate]

Laertes and Roxane go to the Senate to play a game of Hide-and-Seek. There are 100 rooms in the Senate, and Roxane picks one of them and hides there till the game ends. Laertes, at the ...
2
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1answer
28 views

OIES formula for summation not working

I have the following summation: $$F(k)=\sum\limits_{n=1}^k\sum\limits_{d|n}\gcd\left({d},{\frac{n}{d}}\right)$$ At this OEIS link (http://oeis.org/A055155), this exact summation is found. (Credits ...
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7answers
173 views

Prove $1^2-2^2+3^2-4^2+…+(-1)^{k-1}k^2 = (-1)^{k-1}\cdot \frac{k(k+1)}{2}$

I'm trying to solve this problem from Skiena book, "Algorithm design manual". I don't know the answer but it seems like the entity on the R.H.S is the summation for series $1+2+3+..$. However the ...
1
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1answer
73 views

Prove (or disprove) that $ \sum_{n=1}^\infty \frac{4(-1)^n}{1-4n^2} x^n = \frac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} - 2 $ for $ 0<x\leq1$

Just like title said, for $ 0 <x\leq1 $, prove/disprove: $$ \displaystyle \sum_{n=1}^\infty \dfrac{4(-1)^n}{1-4n^2} \cdot x^n \stackrel{?}{=} \dfrac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} - 2 $$ I ...
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1answer
75 views

Closed form for $\sum_{k=1}^n\frac{1}{(2k-1)(2k+1)}$ [closed]

Find the closed form of $$\sum_{k=1}^n\frac{1}{(2k-1)(2k+1)}$$
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0answers
103 views

A combinatorial identity involving generalized harmonic numbers

The $n$-th harmonic number is defined as $$ H_n=\sum_{k=1}^{n}\frac{1}{k}, $$ and the generalized harmonic numbers are defined by $$ H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}. $$ Recently, I have found ...
2
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1answer
67 views

Cesaro summation

Let's consider $\{a_{n} \} -$ a bounded sequence of real numbers. Is it true that $\frac{1}{n} \sum_{k=0}^{n-1}{|a_{k}|^{p}}$ (Cesaro sums) converges or diverges for all $p \geq 1$? (more presicely: ...
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1answer
43 views

$a_{3n} = {-1}/ \sqrt[3]n, \ a_{3n+1} = {-2}/ \sqrt[3]n, \ a_{3n+2}= {3} / \sqrt[3]n$ then $\sum_{n=1}^{\infty} a_n$ converges

Let $a_n$ defined by: $$a_{3n} = \frac{-1}{\sqrt[3]n},\quad a_{3n+1} = \frac{-2}{\sqrt[3]n},\quad a_{3n+2}=\frac{3}{\sqrt[3]n} $$ show that $\sum_{n=1}^{\infty} a_n$ converges. I thought about ...
2
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1answer
54 views

Induction based on sum of $kth$ powers. [duplicate]

It is showable directly by induction that the following are true: $$\sum k = \frac{1}{2}n(n+1)$$ $$\sum k^2 = \frac{1}{6}n(n+1)(2n+1)$$ $$\sum k^3 = \frac{1}{4}n^2(n+1)^2$$ etc. Now, by doing some ...
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0answers
25 views

Symbol for Sequential Subtraction

I was just curious that why there is no symbol for sequential subtraction in maths. This is unlike summation and Multiplication? Each having their respective symbols as $\Sigma $ and $\Pi$, namely.
42
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2answers
1k views

Xmas Greeting 2015

Simplify the expression below into a seasonal greeting using commonly-used symbols in commonly-used formulas in maths and physics. Colours are purely ornamental! $$\large \begin{align} \frac{ ...
6
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2answers
107 views

How to simplify $F(k)=\sum\limits_{n=1}^k\sum\limits_{d|n}\gcd({d},{\frac{n}{d}})$?

I have the following summation: $$F(k)=\sum\limits_{n=1}^k\sum\limits_{d|n}\gcd\left({d},{\frac{n}{d}}\right)$$ This is nearly impossible to compute (using coding) for large numbers, due to the ...
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1answer
37 views

An inequality involving sums and products

I am curious to know whether the following holds or not. If $n_1,n_2,n_3,m_1,m_2$ are positive integers strictly greater than 1 such that $$n_1+n_2+n_3 > m_1 +m_2$$ then $$n_1n_2n_3 \geq m_1m_2.$$ ...
3
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1answer
49 views

Is there a standard notation for a sum (or product) over the elements of a set?

I would like to know if there is a standard notation for the sum/product of a function over the elements of a set. I have used the following notations before: $$\sum_{x\in S}f(x)$$ $$\prod_{x\in ...
3
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1answer
49 views

Confusion regarding a series

I tried much but was unable to find the answer. $$f(x) = \frac{1}{3} + \frac{1 \cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9} + \frac{1\cdot 3\cdot 5\cdot 7}{3\cdot 6\cdot 9\cdot 12} ...
3
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1answer
73 views

Is there a name for this expression?

Is there a commonly accepted name for expression of the form: $$ \sqrt{\sum_i V^2_i + \sum_i\sum_{j\neq i}\kappa_{ij}V_i V_j} $$ where V is a vector and $\kappa$ is a matrix of weights. sorry if my ...
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1answer
40 views

Binomial Coefficients within partial sums

I need to be able to show that: $\sum_{k=i}^{n} {n \choose k} (1-t)^{n-k} t^{k-i} {k \choose i} (1-\tau)^{k-i}$ is equivalent to ${n \choose i} (1-\tau t)^{n-i}$. However I have no idea how to expand ...
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2answers
220 views

Calculate the sum of first $n$ natural numbers taken $k$ at a time

So sum of first $n$ natural numbers taken $1$ at a time is $n\cdot(n+1)/2$ but what about $2,3,\dots,k$ at a time? Is there a general formula? For example, taking 1 at a time $$\sum_{i = 1}^{n} i = ...
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2answers
73 views

Approximating a Harmonic Sum

The infinite sum $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges. However, it is possible to find bounds from some $n$ to another integer $n$. Wolfram alpha is able to give a decimal approximation of the ...
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2answers
123 views

Formula for $\sum \limits_{n=0}^{\infty} \frac{1}{(n+a)!}$

Is there a closed form for the infinite sum $$\sum \limits_{n=0}^{\infty} \frac{1}{(n+a)!} \mathrm{?}$$ where a is an integer greater than or equal to $0$. When $a=0$, the sum is just the series ...
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1answer
38 views

Is $\sum_{x=1}^{\infty} P(X \le x, Y>x)$ not the same as $\sum_{x=2}^{\infty} P(X<x|Y=x)P(Y=x)$?

I'm trying to figure out how to calculate $P(X<Y)$ for discrete random variables, taking values in the positive integers (so excluding 0). I've come up with a few ways. In my consideration the two ...
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1answer
23 views

How to choose rows from a table, such that it adds up to a given vector?

I have a table of food stuffs and their cost and dietary nutrient supply like say vitamins, calories, minerals etc. I want to choose certain food stuffs from the table so that their summation meets ...
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1answer
59 views

Can finite sums of two numbers come arbitrarily close to zero?

Given two real numbers $a$ and $b$, define an $a$-$b$-sum as a finite sum of $a$'s and $b$'s, i.e. a sum: $$m\cdot a + n\cdot b$$ where $m,n$ are non-negative integers. Is there a pair of numbers ...
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2answers
143 views

It is possible to get a closed-form for $1+2^i+3^i+\cdots (N-1)^i$?

Let $i=\sqrt{-1}$ the complex imaginary unit, taking $$arg(2)=0$$ for the definition of the summand $2^i$ in $$1^i+2^i+3^i+\cdots (N-1)^i,$$ as $$2^i=\cos\log 2+ i\sin\log 2,$$ see [1]. Question. ...
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0answers
135 views

Sum of like powers equal to a power

It's not hard to prove that $$(1+2+3+\ldots+n)^2=1^3+2^3+\ldots+n^3$$ ( for example using induction ) A generalization of this is also known : $$(\sum_{d \mid n} \tau(d))^2=\sum_{d \mid n} ...
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1answer
72 views

Double Summation Over all subset of $\{1,2,…n\}$

In Benson's Book "Polynomial In variants of Finite Groups" It is claimed that(Without any proof): $$ j! u_1u_2...u_j =\sum_{I \subseteq \{1,2,...,j\} } (-1)^I (\sum_{i \in I}u_i)^j$$ Where $I$ runs ...
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0answers
92 views

How to compute double summations where the two summands are not independent?

Edit: From the vote counts I see that people want this question closed as it seems unclear what I was asking, so I have tried to word it a bit better to avoid closure. I hope this helps, please ...
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2answers
39 views

Double sum of products of integers up to $n$

Suppose that $S$ is defined by $$ S(n) = \sum_{i=0}^{n} \sum_{j=0}^{i} ij. $$ I'm confused as to how $S(3) = 25$ from this summation. Can anyone expand on it as to how to get the answer?
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28 views

Matsubara sum with general exponent

Matsubara sums of the form $$\sum_{i\omega}\frac{1}{(i\omega-\xi_1)^a}\frac{1}{(i\omega-\xi_2)^a} $$ have closed-form solutions for $a=1,2$. See Wikipedia. Are there also closed-form solutions for ...
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0answers
13 views

Simplifying an expression which includes summation symbols and the cumulative distribution function for the normal

I would like to be able to simplify the expression: $E(Y|\mu,\sigma^2) = \frac{\sum_1^J 1 - 2 \Phi((c_j - \mu)/\sigma) + 2 \Phi((c_j - \mu)/\sigma)^2}{J}$ where $\Phi$ is the cumulative distribution ...
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0answers
52 views

Evaluate $\sum_{n=1}^\infty\frac{2^{-2^n}\cos{(2^n)-2^{-3(2^n)}\cos{(3(2^n))}}}{2^{2^{n+2}}-2^{1-2^{n+1}}\cos{(2^{n+1})}+2}$

I want to find the value of $$\sum_{n=1}^\infty\frac{2^{-2^n}\cos{(2^n)-2^{-3\cdot2^n}\cos{(3\cdot2^n)}}}{2^{2^{n+2}}-2^{1-2^{n+1}}\cos{(2^{n+1})}+2}$$ We have an identity ...
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1answer
114 views

$S1 = 1 + {x^3 \over 3!} + {x^6 \over 6!} + …$

In one of my lecturer's problem sheets we were asked to evaluate the following sums: $$S1 = 1 + {x^3 \over 3!} + {x^6 \over 6!} + \dots $$ $$S2 = {x^1 \over 1!} +{x^4 \over 4!} +{x^7 \over 7!} + ...
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0answers
35 views

How does one generally express a symmetric summation into matrix multiplication?

In the summation ,$$\sum_{i}^{}\sum_{j}A_{ij}X_{i}X_{j}$$ a nice symmetry exists. The final sum of this summation is just \begin{matrix} (A_{11} & A_{12} & A_{13}) X_{1} \\ (A_{21} & ...
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0answers
28 views

Binomial square sum and product

Given $c,n\in\Bbb N$ what is the expression for $$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$ and $$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$ where $x-c<c\leq ...
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1answer
32 views

Einstein Summation with Del Operator

Can someone show explicitly me why $2B_k\nabla B_k = \nabla B^2$ ? Is $B_k\nabla B_k$ just $B_x\nabla B_x+B_y\nabla B_y+B_z\nabla B_z$? But then I end up with nine terms on the LHS and I can't ...
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2answers
48 views

I Know that $\sum_{n=0}^\infty \frac{1}{n}$ Diverges, but what is $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ [duplicate]

We know that $\sum_{n=0}^\infty \frac{1}{n}$ diverges since it is a harmonic series. However, I was recently working on a homework problem where I was given to find if $\sum_{n=1}^{\infty} ...
3
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2answers
76 views

Is there a name for a binomial expansion without coefficients?

I am investigating a problem from George E. Andrews Number Theory (Dover, 1971), discussed previously here: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$ I was led ...
2
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1answer
39 views

Assymptotics of the generalized harmonic number $H_{n,r}$ for $r < 1$

The $H_{n,r}$ generalized harmonic number is defined as: $$H_{n,r} = \sum_{k=1}^{n} \frac{1}{k^r}$$ I'm interested in the growth of $H_{n,r}$ as a function of $n$, for a fixed $r\in[0,1]$. For ...
4
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1answer
142 views

Tricky proof that the weighted average is a better estimate than the un-weighted average:

The following is a word for word copy of a tough question and the solution to it. I have marked $\color{red}{\mathrm{red}}$ the parts of the solution for which I do not understand and the parts marked ...
1
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0answers
10 views

Convergence of a Double Summation solution to Laplace's Equation

For a cube of side length $a$ with 2 opposite sides held at the same potential $V$, the potential at the center of the cube can be expressed in series form as And I am trying to show that this ...
3
votes
1answer
71 views

$H_k$ summability of a sequence implies its Abel summability to the same sum.

Let $\sigma_n^{(k)}=\frac{1}{n+1}\sum_{j=0}^{n}\sigma_j^{(k-1)}$ and $\sigma_n^{(1)}=\frac{1}{n+1}\sum_{j=0}^{n}s_j.$ If $\lim_{n\to \infty}\sigma_n^{(k)}=L$ we call the sequence $(s_n)$ is summable ...
3
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2answers
81 views

Prove: $\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1)$

Prove: $\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1)$ What method to use in order to find the closed form of summation ...
2
votes
2answers
52 views

How to recognize / convert a tricky limit of an infinite series as a Riemann integral?

Edit: I've modified the sums and integrals below into convergent sums and integrals, but my questions are still the same - how can I convert sums into integrals legitimately? As far as I know, the ...
0
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1answer
42 views

How to write sum notation for an array of 2D points

What is a correct way to write using sigma notation for a problem involving an array of 2-dimensional points. Say I have 2 arrays, $P_{e}$ and $P_{a}$, both containing $N$ elements. $P_{e}$ represents ...
9
votes
4answers
150 views

What does the false infinite sum of a series mean?

For any geometric series with |$r$| < 1 , I know that $$\sum_{k=1}^{∞} ar^{k-1} =\frac{a}{1-r}$$ But if |$r$| > 1 and you try to use the formula, you'll get a weird answer. For instance: ...
2
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1answer
95 views

Summation relating factorial and cosine

How to simplify \begin{align*} \sum_{k=0}^{\infty}\left(-1\right)^{k}\frac{\left(2k\right)!}{4^{k}\left(k!\right)^{2}}\cos\left(kx\right) \end{align*} for $0\leq x <\pi$ ? I don't even know where ...