# Tagged Questions

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 ...
62 views

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 ...
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 $$n_1n_2n_3 \geq m_1m_2.$$ ...
49 views

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 ...
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 ...
220 views

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 ...
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 ...
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?
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 ...
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 ...
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 ...
In one of my lecturer's problem sheets we were asked to evaluate the following sums: $$S1 = 1 + {x^3 \over 3!} + {x^6 \over 6!} + \dots$$ $$S2 = {x^1 \over 1!} +{x^4 \over 4!} +{x^7 \over 7!} + ... 0answers 35 views ### How does one generally express a symmetric summation into matrix multiplication? In the summation ,$$\sum_{i}^{}\sum_{j}A_{ij}X_{i}X_{j}$$a nice symmetry exists. The final sum of this summation is just \begin{matrix} (A_{11} & A_{12} & A_{13}) X_{1} \\ (A_{21} & ... 0answers 28 views ### Binomial square sum and product Given c,n\in\Bbb N what is the expression for$$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$$where x-c<c\leq ... 1answer 32 views ### Einstein Summation with Del Operator Can someone show explicitly me why 2B_k\nabla B_k = \nabla B^2 ? Is B_k\nabla B_k just B_x\nabla B_x+B_y\nabla B_y+B_z\nabla B_z? But then I end up with nine terms on the LHS and I can't ... 2answers 48 views ### I Know that \sum_{n=0}^\infty \frac{1}{n} Diverges, but what is \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} [duplicate] We know that \sum_{n=0}^\infty \frac{1}{n} diverges since it is a harmonic series. However, I was recently working on a homework problem where I was given to find if \sum_{n=1}^{\infty} ... 2answers 76 views ### Is there a name for a binomial expansion without coefficients? I am investigating a problem from George E. Andrews Number Theory (Dover, 1971), discussed previously here: Induction Proof that x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}) I was led ... 1answer 39 views ### Assymptotics of the generalized harmonic number H_{n,r} for r < 1 The H_{n,r} generalized harmonic number is defined as:$$H_{n,r} = \sum_{k=1}^{n} \frac{1}{k^r}$$I'm interested in the growth of H_{n,r} as a function of n, for a fixed r\in[0,1]. For ... 1answer 142 views ### Tricky proof that the weighted average is a better estimate than the un-weighted average: The following is a word for word copy of a tough question and the solution to it. I have marked \color{red}{\mathrm{red}} the parts of the solution for which I do not understand and the parts marked ... 0answers 10 views ### Convergence of a Double Summation solution to Laplace's Equation For a cube of side length a with 2 opposite sides held at the same potential V, the potential at the center of the cube can be expressed in series form as And I am trying to show that this ... 1answer 71 views ### H_k summability of a sequence implies its Abel summability to the same sum. Let \sigma_n^{(k)}=\frac{1}{n+1}\sum_{j=0}^{n}\sigma_j^{(k-1)} and \sigma_n^{(1)}=\frac{1}{n+1}\sum_{j=0}^{n}s_j. If \lim_{n\to \infty}\sigma_n^{(k)}=L we call the sequence (s_n) is summable ... 2answers 81 views ### Prove: \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1) Prove: \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1) What method to use in order to find the closed form of summation ... 2answers 52 views ### How to recognize / convert a tricky limit of an infinite series as a Riemann integral? Edit: I've modified the sums and integrals below into convergent sums and integrals, but my questions are still the same - how can I convert sums into integrals legitimately? As far as I know, the ... 1answer 42 views ### How to write sum notation for an array of 2D points What is a correct way to write using sigma notation for a problem involving an array of 2-dimensional points. Say I have 2 arrays, P_{e} and P_{a}, both containing N elements. P_{e} represents ... 4answers 150 views ### What does the false infinite sum of a series mean? For any geometric series with |r| < 1 , I know that$$\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: ...
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 ...