# Tagged Questions

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### Need to understand this summation with max notation

Firstly, apologies needed for my math description if it does not sound right. I have come across a paper where I saw a summation notation with a max function in it which I am little confused to ...
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### Another Hockey Stick Identity

I know this question has been asked before and has been answered here and here. I have a slightly different formulation of the Hockey Stick Identity and would like some help with a combinatorial ...
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### Evaluate $\int_1^N \frac{-3N+6t-3}{t^3(N-t+1)^4}dt$ when $N=3$ or $N=5$

Let the Cauchy product $$(\zeta(3))^2=\sum_{n=1}^\infty c_n,$$ where $$c_n=\sum_{k=1}^n\frac{1}{k^3(n-k+1)^3},$$ and $\zeta(3)$ is the Apèry constant. Taking $f(x)=\frac{1}{x^3(N-x+1)^3}$ in Abel's ...
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### Matlab sum is wrong: Double symsum gives incorrect result

I'm trying to calculate the double sum $$\frac{1}{10} \sum_{x=1}^{10} \left( \frac{1}{x} \sum_{n=0}^{floor(log_{10}x)} 10^n \right).$$ In MATLAB, my result is ...
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### Finding $\sum_{j=m}^{n}\frac{a^{j-m}}{N-j}$

How can we tackle $$\sum_{j=m}^{n}\frac{a^{j-m}}{N-j}$$ when $0<m<n<N$. I have been using Euler-Maclaurin sum to change this object to some integral but it gets a bit messy. I appreciate any ...
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### Series with Markov Chains Probabilities

Notation For each $t \in \mathbb{N}$, let $h_t \in H$ be a random variable that follows a Markov chain, and $h^t \equiv \{h_0,h_1,\dots,h_t\} \in H^t$. Let $\Pi(h^{t})$ be the probability that a ...
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### How can you simplify this expression for amount of triangles?

I was given a question as a challenge were I was supposed to find a formula to find how many triangles there are when you draw $n$ and $m$ amount of lines from points $N$ and $M$ to the opposite sides....
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### Do not understand algebra technique used to computer summation

I am going through a practice exam for my Discrete Mathematics class and do not understand the algebra used in the following summation computation. Summation to compute: Answer: What I don't ...
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### Limit of a series with a lot of dependencies

Let $n \rightarrow \infty$ and consider $$\sum_{x=\lfloor \log^6(n)\rfloor}^{\lceil \frac{n}{\log^2(n)}\rceil} \left(\frac{n}{\log^2(n) x} e^{-\frac{\log^{16}(n)}{n}}\right)^x$$ Do we know anything ...
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### double summation notation

In a paper I am studying, the author writes $$\sum_{{i=1}\atop {k=1}}^{N+1} C_i \eta_k$$ How are the two indices to be interpreted? In other words, how would this expression be written using sigma ...
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### How to prove this quasi-geometric trigonometric series identity without induction

$$\frac{2}{\sin{x}}\sum_{r=1}^{n-1} \sin{rx}\cos{[(n-r)y]} \equiv \frac{\cos{(nx)}-\cos{(ny)}}{\cos{x}-\cos{y}} - \frac{\sin{(nx)}}{\sin{x}}$$ The identity can be tediously proven using the Axiom of ...
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### Sum of all distinct numbers made

Question: Find the sum of all distinct four digit numbers that can be formed using the digits 1; 2; 3; 4; and 5, each digit appearing at most once. I have no clue as to where to begin this question. ...
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### $T$ can be $\infty$ with positive probability

From Williams' Probability with Martingales How exactly do we know $T$ can be $\infty$ with positive probability or $$P(T = \infty) > 0 \text{ ?}$$ I'm guessing that that means there ...
### Sum of $1/n^k$ of the first $\log P$ numbers
In a Udacity course I'm told the following: $\sum_{i=1}^{\log_2 (P)} 1/2^i = (P-1) /P$ I've checked that it's true by entering it into Wolfram Alpha: https://www.wolframalpha.com/input/?i=sum+1%2F2^...