# All Questions

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### doubt with sequence and series question

show that the series ((1+x)/(1-x))^3=1+ summation n=1 to infinity (4n^2+2)x^n I tried with the partial frationaising the expression that gives me -6/(x-1) -12/(x-1)^2 -8/(x-1)^3 -1 how to ...
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### System of linear diophantine modular inequalities

How can we best find a numerical solution to a system of $m\ge2$ linear diophantine modular inequalities $$\big((a^j x+b_j)\bmod n\big)<c\;\text{ for }1\le j\le m$$ where $x$ is the only unknown, ...
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### Powers of adjacency matrix doesn't seem to correspond to observed number of paths on graph

I would really appreciate some help on this! $A^n$ represents $n^{th}$ power of the adjacency matrix of a graph. I keep reading that the $A^n_{ij}$ entry equals "the number of paths of length n ...
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### Integers Placed On A Circle

My problem is such: On a circle there are $9$ distinct positive integers aranced in such a way that the product of two non-adjacent numbers in the circle is a multiple of $n$ and the product of any ...
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There is this proof for the integral of convolution between two functions: \begin{align}\int_{-\infty}^{\infty} (f*g)(x)dx&=\int_{-\infty}^{\infty}\left [ ... 2answers 40 views ### Why does the radius come before the angle? Based on my understanding, when delineating two variables (for a coordinate system or otherwise) convention is to label the 'independent variable' first, then the 'dependent variable'. So for a ... 2answers 40 views ### What is the probability of sinking ships in a simplified game of battleship? Consider a a simplified game of battleship. We are given a 4x4 board on which we can place 2 pieces. One destroyer which is a 1 × 2 squares and a submarine that is 1 × 3 squares ￼. The pieces are ... 1answer 57 views ### Philosophers who became mathematicians — how did they do it? And who were they? (I hope this is not too personal. If you want to get to the point scroll down to the end, where my questions are.) I'm a philosopher who's been -- gradually -- coming around to mathematics. I have ... 1answer 40 views ### If (a,b)=1 and p \mid a^{2}+b^{2} why can one assume that |a| < \frac{p}{2} There is a part of Euler's infinite descent proof I can't seam to get; If (a,b)=1 and p \mid a^2+b^2 why can one assume that |a| < \frac{p}{2} and |b| < \frac{p}{2} ? 1answer 24 views ### P(\min(X_1,\dots,X_n) > t) = P(X_1>t,\dots, X_n>t) One step in the my solutions book shows... P(\min(X_1,\ldots,X_n) > t) = P(X_1>t, \ldots, X_n>t), where X_1, \ldots, X_n  are independent and X_j \sim \mathrm{Expo}(\lambda)  Why is ... 0answers 24 views ### Finding the sum [on hold] How to find the sum of \sum_{n=1}^{\infty}\frac{(-1)^n}{2^nn} ? 3answers 25 views ### If T : F^{2 \times 2} \to F^{2\times 2} is T(A) = PA for some fixed 2 \times 2 matrix P, why is \operatorname{tr} T = 2\operatorname{tr} P? I am asked to prove that if T is a linear operator on the space of 2 \times 2 matrices over a field F such that T(A) = PA for some fixed 2 \times 2 matrix P, then ... 0answers 24 views ### about the ratio of the coefficients [duplicate] Let f be holomorphic on an open disk containing the unit circle, except in a pole w on the unit circle. Let \displaystyle \sum_{k=0}^{\infty} a_n z^n be its expansion. Show that ... 4answers 55 views ### How to prove that the sum of binomials equals \begin{pmatrix}2n\\n\end{pmatrix} [duplicate] I've stumbled upon this lemma a few times in my textbook:\sum_{k=0}^{n}\begin{pmatrix}n\\k\end{pmatrix}=\begin{pmatrix}2n\\n\end{pmatrix}$$I've been trying to prove it, but I simply can't seem to ... 0answers 33 views ### How to simplify this inequality I have the following inequality where i, N and p are constants, j is a variable and p_j is the chance that 'event' j is happening:$$i\geq -pi+((1-p)\cdot \sum ^N _{j=0}(j\cdot p_j))+\sum ...
In JavaScript, the largest odd positive number representable is $2^{53}-1$. All integers between 1 and $2^{53}-1$ can be represented without loss of precision. How many prime numbers can be ...