Questions tagged [summation]

1,996 questions
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Which cycles are possible by repeated summation of the cubes of the digits of a number?

Here : Digital root with squared digits the possible cycles of repeated summation of the squares of the digits of a number are mentioned. What about cubes ? $1$ , $153$ , $370$ , $371$ and $407$ ...
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Calculate the sum $\sum\limits_{i=1}^n \sum\limits_{j=1}^n a_i \mod a_j$ in $\mathbb{Z}.$

Calculate the sum $$\sum_{i=1}^n \sum_{j=1}^n a_i \mod a_j$$ for some integer $a_i \in \mathbb{Z}.$ If we calculated the usual sum $$\sum_{i=1}^n \sum_{j=1}^n a_i a_j$$ then it would be easy ...
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“Weighted” Summation

Since already two utents have misunderstood the question, I'm pointing it out: I've already proved the first problem, I'm using it just as an example and as motivation to the question, which is the ...
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Linear algebra and unknown sum limits

I have been struggling with the this one. Let's assume that: $N$ is a large integer $\epsilon$ is real and $0<\epsilon<1$ $\rho$ is real, $0<\rho<1$ and $\rho N$ is an integer $\alpha_i$ ...
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There are $N+1$ urns, in $i$th urn there are $i$ white balls and $N-i$ red balls, $n$ draws from a rand urn gave red balls, prob next ball also red?

The task: There are $N+1$ urns. In the $i$th urn there are $i$ white balls and $N-i$ red balls for $i=0,\ldots,N$. We choose a random urn and then we choose $n$ times a random ball from this urn ...
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Partial sum involving harmonic numbers

QUESTION: I need to know how to compute the partial sum $$\sum_{k=1}^n \frac{H_{k+1}^2-H_{k+1}^{(2)}}{k+2}$$ in terms of the generalized harmonic numbers $H_n^{(m)}$. CONTEXT: This problem arose ...
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Find a sum of a convergent series

Let $x_n$ be a sequence that is given by the following recursive formula: $x_{n+1} = x_n^2 - x_n +1$, where $x_1=a \gt 1$. Find: $$\sum_{n=1}^{\infty} \frac{1}{x_n}$$ Not sure really how to ...
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Finding basis of coefficients satisfying system of linear equations

Setup: Let $C_{i,j}$ denote coefficients which are symmetric in their indices---i.e., $C_{ij}=C_{ji}$. These coefficients are required to satisfy the following relation for all $m\in \mathbb{Z}^+$ ...
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Sum of all permutations of degree $n$

Find the sum of the permutations of degree $n$. For example, $n = 3$ would give: $123 + 132 + 213 + 231 + 321 + 312 = 1332$ I posed this problem to myself, and was able to solve it (I have put my ...
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Evaluating the expected value of $\log(g(x))$.

I am trying to evaluate the expected value of $\log(x+r)$ with respect to a negative binomial distribution. In other words, I am searching for a closed form or approximation to the quantity \begin{...
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