# Tagged Questions

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### Distribution of distances between points with complete spatial randomness

I'm trying to compute the probability of the distances between points on a 2D domain that have complete spatial randomness (CSR). From this wikipedia page on CSR, the probability of locating the $N$...
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### Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$

As the title says I need to prove the following by induction: $$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$ When trying to prove that P(n+1) is true if P(n) is, then I ...
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### Factorial Proof by Induction Question? [duplicate]

$\text{Use the PMI to prove the following for all natural numbers n.}$ $\frac{1}{2!} + \frac{2}{3!} + \cdot \cdot \cdot + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}$ So for this question I get ...
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### Is difference of two consecutive sums of consecutive integers (of the same length) always square?

I am an amateur who has been pondering the following question. If there is a name for this or more information about anyone who has postulated this before, I would be interested about reading up on it....
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### Summation over simplices in higher dimensions

Let $l\ge 1$ and $s\ge 1$ be integers. Define $\vec{a} := \left(a_j\right)_{j=0}^s$. The following set: \Delta^{(s)}_l := \left\{ \vec{a} \quad \left| \quad \sum\limits_{\eta=0}^s a_\...
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### The 1000th partial sum of the series $\sum 1/n^2$ is less than 2 [closed]

Can anyone help me with this problem: Prove: $$1/1^2 + 1/2^2 +1/3^2 +\dots +1/1000^2 <2$$
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### What is $\sum_{i=1}^{n}\frac{F_i}{i}$?

Mathematica is able to evaluate the summation $\sum_{i=1}^{n}\frac{F_i}{i}$ in terms of the Lerch transcendent. It is natural to consider whether or not this summation can be expressed in a more ...
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### Reducing an indicator function summation into a simpler form.

Context I am attempting to reduce the space I need to store in an array in a program. I have made it so that the indices are always sorted. There are no indices where they are equal, and no indices ...
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### sum of falling factorial $\sum_{k=0}^{n-1}\frac{n!}{(k+1)!}a^{n-k-1}$

I want to compute $\sum_{k=0}^{n-1}\frac{n!}{(k+1)!}a^{n-k-1}$. I note that it is similar to a generating function. The coefficients are falling factorials. Can I simplify it? Thanks!
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### How was the integral for Zeta Function created

How was the zeta function integrated from $$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ To $$\zeta(s) = \frac{1}{\Gamma (s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}dx$$ I've tried googling ...
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### Identity involving the Catalan numbers and binomial coefficients

Let $C_k := \frac{1}{k + 1} \binom{2k}{k}$ be the $k$-th Catalan number and let $K$ be a positive integer. I am looking for an identity or simplification of \sum_{k = 0}^K C_k \...
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### Function $f: \mathbb{Z} \to \mathbb{Z}^n$ related to $\sum_{k=1}^{x} k^n$.

The sequence $\{a_0,a_1,...a_x\}$ has closed form $a_n=\sum_{i=0}^{\infty} \Delta^i(0) {n \choose i}$ where $\Delta a_n$ denotes the operation mapping $a_n$ to $a_{n+1}-a_n$ and $\Delta^i(0)$ is ...
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### How can this double summation be solved?

I have to calculate the following expectation $$\mathbb{E}\left[\left(\frac1M\sum\limits_{i=1}^MX(i-n_1-M)\right)\left(\frac1M\sum\limits_{j=1}^MX(j-n_2-M)\right)\right]$$ where $M$, $n_1$ and $n_2$ ...
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### Is there a closed formula for this summation?

I have the summation $$\sum\limits_{i=1}^n \frac{1}{i^2}$$ And I don't know how to find a closed formula for it. Any ideas?
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### Fibonacci summation

Can anyone help me to prove the following relation. $$\sum_{k=1}^{\infty} \frac{F_{2k}H^{(2)}_{k-1}}{k^2\binom{2k}{k}}=\frac{2\pi^4}{375\sqrt{5}}$$ I was studying recently about Fibonacci and ...
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### Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem

I'm trying to prove that $$\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$$ with the Binomial Theorem. I know that the B.T. states that (x + y)^n = ...
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### Riemann Zeta Function, Stirling's Numbers, and Infinite Series

A while back I was able to prove the following identity, $$\sum_{k=1}^{\infty}\frac{\Gamma(k+r)}{\Gamma(k)(k+r)^s}=\sum_{k=0}^{r}s(r+1,r+1-k)\zeta(s-r+k)$$ where $s(k,n)$ are the Sterling numbers of ...
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### Has this summation a specific result?

I need to calculate this integral $$\sum\limits_{k=1}^\infty\int\limits_I \frac{\lambda^k}{(k-1)!}t_k^{k-1}e^{-\lambda t_k} dt_k$$ where $I=(a,b)$. Someone told me that summation equals $\lambda$, ...
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### What should be expected values of weights in statistics (specifically a set of biweight weightings)

I am working on a project that involves taking the biweight sample variance of a velocity dataset. This is defined as $\sigma_{BI}^2 = N\dfrac{\sum_{|u_i|<1}(1-u_i^2)^4(v_i-\bar{v})}{D(D-1)}$ ...
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### Reducing Binomial Summation [closed]

How can I reduce this summation into this $$\frac{1}{2}(1+\left(1/3\right)^{50})$$ The problem comes from the 1992 AHSME Test (problem 29)
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### Convergence of $\sum_{n\in\mathbb{N}_{>0}} f(n)\mathrm{log}(n)$

A function $f(n)$ has the following conditions: $$f(n),n\in\mathbb{N}_{>0}$$ $$f(n)\in[0,1]$$ $$\sum_{n\in\mathbb{N}_{>0}} f(n)=1$$ Does the following sum always converge? Or does a ...
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### How to find $\sum_{A \subset S} (\min A)$ and $\sum_{A \subset S} (\max A)$ if $S=\{1,2,…,n\}$?

Here, $\min A$ and $\max A$ denote the minimum and maximum element respectively of the set $A$. So I have to calculate how many subsets of S have min/max element $1$, how many subsets have min/max ...
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### Pattern with the the tetration of summations.

While dealing with a question with finding an explicit form for a sequence I noticed something: $$\sum_{x_0=0}^{n-1} 1=\frac{n}{1!}$$ $$\sum_{x_0=0}^{n-1} \sum_{x_1=0}^{x_0-1} 1=\frac{n(n-1)}{2!}$$ ...
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### If $(A_n)_{n\in \mathbb{N}}$ is an open set in $\mathbb{R}^2$ how do I find an example where $\bigcap\limits_{n=1}^\infty A_n$ also is open?

If $(A_n)$ is an open set in $\mathbb{R}^2$ for all $n\in \mathbb{N}$. How can I find an example where $\bigcap\limits_{n=1}^\infty A_n$ also is open? I'm new to this concept, so any leads would be ...
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### logarithm of a sum or addition

I search a general rule for calculating the logarithm of a sum or addition. I know that $$\ln{(a+b)}=\ln{\left(a\left(1+\frac b a\right)\right)}=\ln{(a)}+\ln{\left(1+\frac b a\right)}$$ but when the ...
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### How to calculate $x=\sum _{ i=0 }^{ \infty } \left( i+1 \right) \cdot \left( \frac { 5 }{ 6 } \right) ^{ i }$ [duplicate]

I was trying to find the value of x in the following equation. $$x=\sum _{ i=0 }^{ \infty } \left( i+1 \right) \cdot \left( \frac { 5 }{ 6 } \right) ^{ i }$$ In a computer simulation, I found that ...
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### What's wrong with this formula for the dot product of a vector and a matrix acting on that vector?

Suppose we have an $n \times n$ matrix $M$ and a vector $v$. I want to find an explicit formula for $v \cdot Mv$. I begin by saying $$v \cdot Mv = \sum_{i=1}^{n} v_i(Mv)_i$$ and since $Mv$ is given by ...
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### Sum of nth powers of Fibonacci numbers

Is a closed form for $$\sum_{i=1}^n{F_i^k}$$ (where $F_i$ is the $i^{th}$ Fibonacci number and $k$ is constant) known?
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### Can I calculate a fractional sum with functional equations and/or infinite series?

I was wondering if I can calculate fractional sums (non-integer sums) with functional equations in the following manner $$f(x):=\sum_{n=1}^xg(n)$$ $$f(x)=g(x)+f(x-1)\tag1$$ $(1)$ most certainly ...
Consider the "equation" $$\frac{1}{a_n}\sum_{k=1}^n ka_k = \mathcal{O}\left(\frac{n^2}{\log n}\right).\tag{1}\label{eq:conjec}$$ Does there exist some monotonically ...