# All Questions

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### Find the formula of a sequence using generating functions

The given sequence is $\sum_{k=1}^{n}kx^{k}$. How can I do this? I think I should start with the already known generating function: $A(x) = \sum a^{n}x^{n} = \frac{1}{1 - ax}$ And then I should do ...
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### Unexpected approximations which have led to important mathematical discoveries

On a regular basis, one sees at MSE approximate numerology questions like Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, ...
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How to verify whether $C_{00}=\{(x_n):\text{All but finitely many terms are }0\}$ is complete with respect to $p$-norm given by $$\|(x_n)\|_p=\left(\sum_{n=1}^\infty|x_n|^p\right)^{1/p}$$ where $1\le ... 2answers 821 views ### Calculate distance from plane to parallel plane in O using vector and normal I'm trying to figure out what's the best method to get the distance between two planes where i have the normalized vector of the plane and a point in the plane. What I want to do is to create a ... 1answer 189 views ### Finding the value of n if$2^{200}-2^{192}31 +2^n$is perfect square Problem : Finding the value of n if$2^{200}-2^{192}31 +2^n$is perfect square Solution :$2^{200}-2^{192}31 +2^n = 2^{192}(2^8-31)+2^n$=$2^{192}(256-31)+2^n = 2^{192}(225)+2^n\therefore$... 0answers 140 views ### Differential calculus on Banach space I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly. Problem Given the ... 1answer 108 views ### Does there exist such a sequence? Does there exist a infinite sequence of positive integers$a_n$. Such that$ ((n|a_n) | \forall n)$and$\left(\sum_{n=1}^\infty \frac {1}{a_n} =1\right)$, what if we replace 1 with a positive real ... 2answers 64 views ### Finding unknown when$0/0$Here Inderteminate(variable) written that $$"x={a-2\over 3-b}$$ while for b = 3 the problem either has no solution at all, or, if also a = 2, it admits any value for x as solution." My question ... 2answers 78 views ### Let$G$be a planar (simple) graph with$m (≥2)$edges and$f$faces. Prove that$f ≤ \frac{2}{3} m$. I am a first year math student, and I'm studying some exercises for my graph theory exam. This exercise I didn't understand very well: Let$G$be a planar (simple) graph with$m (≥2)$edges and ... 1answer 627 views ### Expansion of power series for$\frac{\ln(1-x)}{1+x}$My Problem is to expand$f(x)=\dfrac{\ln(1-x)}{1+x}$into a power series. My Approach: from looking onto the Graphs of this function, i know, for rising x the y is falling towards Zero, without ... 1answer 173 views ### Antichain and predense set I am trying to prove$A$is a maximal antichain i ff$A$is predense. It is problem 16 from link here. If$A$is maximal antichain then is easy to prove that$A$is predense. I stuck on proof ... 2answers 313 views ### Riemann surface arising as a quotient of the upper half-plane. Let$H$be the upper half-plane$\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real$\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote$\Gamma$the ... 1answer 138 views ### Why does$ \frac{1}{2\pi i} \int_{\gamma} \frac{f'(z)}{f(z)} dz = \eta( f \circ \gamma, 0) $($ \gamma $is a regular closed curve) My teacher wrote the above equation on the board the other day and acted like it was obvious. Here,$ \eta $is the winding number of$ f \circ \gamma $. Here is the justification he gave: $$... 1answer 202 views ### Volume of a hypersphere We know that the area of a circle (2-D) =\pi r^{2} and the volume of a sphere (3-D)= \dfrac{4}{3}\pi r^{3}. Question:What is the "volume"(or whatever that is called) of a n-dimensional sphere? ... 1answer 49 views ### Elimination of tickets There are 8 tickets numbered from 1 to 8. They are randomly paired, and larger numbered ticket of each pair is eliminated, leaving only four tickets (the smaller ones in each pair). This process is ... 2answers 70 views ### How to show \sum_{d\mid n} \frac{\mu^2(d)}{d} =\prod_{p|n} \left(1+\frac{1}{p}\right)? how to prove:$$\sum_{d\mid n} \frac{\mu^2(d)}{d} =\prod_{p|n} \left(1+\frac{1}{p}\right)$$\mu : \Bbb N\rightarrow \Bbb R \mu(1)=1 \mu(n)= \begin{cases} 0 &,\;\;\; \text{if \,n\, is ... 2answers 3k views ### Are regular languages necessarily deterministic context-free languages? The original problem Suppose M is DCFL (Deterministic Context Free Language) and N is a regular language. Answer the following questions and justify your answers. a) Is M-N necessarily context-free? ... 1answer 239 views ### About Henstock integrable vector-valued function In what follows, X is a Hausdorff locally convex topological vector space over the reals whose topology is generated by a family P of all continuous seminorms on X. We consider the following ... 1answer 37 views ### Number of Jordan cells size j+1 Could you tell me how to prove that the number of Jordan cells of size j+1 is equal to \dim V^j/V^{j-1}(quotient space)? V^j = \ker (f-\lambda)^j I know that for U_1, U_2, ..., U_s - Jordan ... 3answers 310 views ### Book recommendation for associative algebras Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment. ... 1answer 540 views ### A ring isomorphic to its finite polynomial rings but not to its infinite one. I was messing with the ring k[x_1,\dots,x_n,\dots] of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious: ... 2answers 58 views ### Proof: let f:A \to B with f bijective, then f{\restriction_{C} }: C \to B is bijective I need the proof of following: "let f:A \to B with f bijective, then f{\restriction_{C} }: C \to B is bijective" Thanks in advance 0answers 111 views ### Generators in p-groups Let G be a non-abelian finite p-group such that G/[G,G] is elementary abelian group of order at least p^3. Let S=\{\bar{x_1},\bar{x_2},\bar{x}_3,\cdots,\bar{x}_n\} be independent generators ... 1answer 69 views ### how to find n| 3^n+1 this a problem in: Waclaw Sierpinski 250 problems in elementary number theory 1971 if n\in \Bbb N , n \gt1 ,how to find :$$n\mid3^n+1$$the solution of book is : There is only one such odd ... 2answers 381 views ### is \sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right)<\infty? I don't know why but I'm having a hard time determining whether this series$$ \sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right) $$converges to a real limit. ... 2answers 980 views ### Having trouble understand the proof of Rouché's Theorem I am trying to understand this proof of Rouché's theorem, but I am missing the logic of the last and most crucial step. Here are the assumptions: Suppose that f and g are analytic inside ... 1answer 35 views ### Partitions of an interval and convergence of nets Let \mathscr{T} be the set of partitions \tau = (\tau_0 = 0 < \tau_1 < \dots < \tau_N = 1) of the interval [0,1] (where N is not fixed). This becomes a directed set by setting \tau ... 1answer 213 views ### Canonical Markov Process Let X be a canonical, right-continuous Markov process with values in a Polish state space E, equipped with Borel-\sigma-algebra \mathcal{E} and we assume that t\rightarrow E_{X_{t}}f(X_{s}) ... 0answers 109 views ### (Basic) question regarding Einstein-Hilbert-functional / total scalar curvature I got a question regarding the total scalar curvature / Einstein-Hilbert-functional. I found it in Peter Topping's book "Lectures on the Ricci flow" (you can download it for free here: ... 3answers 163 views ### How does the differential df act on an element of T_pM? Let f be a smooth real valued function on a smooth manifold M. The differential of f is the covector field df defined by$$df_p(v) = v(f)$$where v \in T_pM and where we are now thinking ... 3answers 234 views ### Find the inverse Laplace transform of f(t) = \int_t^\infty \frac{e^{ - u}}{u}du Find the inverse Laplace transform of the integral:$$f(t) = \int_t^\infty {\frac{{{e^{ - u}}}}{u}du} $$If the integral:$$f(t) = \int_0^\infty {\frac{{{e^{ - u}}}}{u}du} $$I had done. However the ... 1answer 111 views ### Understanding topological and manifold boundaries on the real line Let M be the subset [0,1) ∪ {2} of the real line. Find its topological boundary \mathrm{bd}(M) and its manifold boundary \partial M. I know that to find the topological boundary, I ... 0answers 99 views ### Girsanov kernel moments Let Z_t=e^{\int_0^tq_tdB_t-\frac{1}{2}\int_0^tq^2_tdt}, where (q_t)_{t\geq0} is a predictable process, and (B_t)_{t\geq0} a \mathbb{P}-Brownian motion. In particular, Novikov's condition ... 1answer 138 views ### Showing that \int_{0}^{\pi/2}\frac{1+\sin(x)-\sin^2x}{e^{\sin(x)}+\cos(x)}=\log 2 I don't see how to prove that$$\int\limits_{0}^{\pi/2}\frac{1+\sin(x)-\sin^2(x)}{e^{\sin(x)}+\cos(x)}\,\mathrm dx=\log 2$$Integrating by parts is of no help. The same with variable change. Should I ... 0answers 93 views ### Proof of Lie theorem on solvable Lie algebra I am reading a book of Helgason. As you know, solvable Lie algebra g \subset V= {\bf C}^n have a nonzero v such that v is an eigenvector of any element of g. I can follow the proof in ... 2answers 286 views ### Conditions that Roots of a Polynomial be Less than Unity Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ... 1answer 125 views ### The infinity version of Blumenthal's 0-1 law Blumenthal's 0-1 law states that on the space of continuous maps with domain [0, \infty) with the appropriate (Wiener) measure making the coordinate maps Brownian motions starting at x, any event ... 3answers 224 views ### Is \sqrt{-1} positive or negative? Do the concept of positive or negative make sense in this case? I remember that \mathbb{R}^2 has four quadrants thus ordered pairs of numbers could be (+,+),(+,-),(-,-)(-,+), I presume that ... 1answer 107 views ### xy itself square in this particular logic I would like to know the solution or procedure to find the exact analysis/solution of one of my observation. let x = a^2 and y = b^2, then can we express xy (concatenation of x and y) as ... 3answers 65 views ### A problem on recurrence relation Consider the sequence$$a_n = a_{n-1} a_{n-2} +n$$for n \geq 2, with a_0 = 1 and a_1 = 1. Is a_{2011} odd? By writing all the terms of the sequence I see that a_n is odd when n is odd ... 2answers 489 views ### Finding \lim\limits_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n} for continuous f:[0,1]\to[0,\infty) [duplicate] Find$$\lim_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$$if f:[0,1]\rightarrow(0,\infty) is a continuous function. My attempt: Say f(x) has a max. value M. ... 2answers 553 views ### expected number of edges in a random graph If we have a random graph G \in g(n,\frac{1}{2}) how do we show that the expected number of edges is \frac{1}{2} {{n}\choose{2}} Thanks in advance 1answer 141 views ### Rice’s theorem and recursion theorem Prove Rice’s theorem using recursion theorem. I need some hints as to what must be done about it. Please use Davis' book notation: Computability, Complexity, and Languages, Second Edition: ... 1answer 604 views ### Prove a non-empty subset is closed in an inner product space I hope someone would be able to help me with the finer details of this proof. Problem: M is a non-empty set in an Inner Product Space (IPS) X. I need to show that the annihilator of M which is ... 1answer 363 views ### Strength of attraction of fixed points Consider a smooth map f: \mathbb{R} \rightarrow \mathbb{R} with an attracting fixed point F. Then, we have if f'(F) \ne 0, F is a "simple" attracting fixed point, if f'(F) = 0, F is a ... 2answers 211 views ### Are hyperoperators primitive recursive? I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related ... 0answers 97 views ### How to predict order of a set of fractional differential equations? I have a set of differential equations of the form:$$\frac{dv}{dt} = a[b-c*m-d*n-e*h]\frac{dm}{dt} = p(v)\frac{dn}{dt} = q(v)\frac{dh}{dt} = r(v)$$Using fde12 in MATLAB I can ... 1answer 92 views ### uniform convergence of analytic function becomes analytic in a region Suppose that,$G$is a region in the complex plane and$f_n : G \to \mathbb{C}$is analytic for each$n\ge1$. Suppose further that, the sequence {$f_n$} converges uniformly on$G$to a function$f:G ...
Let $F$ be a field, $n$ a positive integer, and $V$ be the space of $n \times n$ matrices over $F$. If $A$ is a fixed $n \times n$ matrix over F, $T_A(B) = AB - BA$. Consider the family of linear ...
### Normal form of a vector field in $\mathbb {R}^4$.
EDIT (In responce to xpaul's answer): I'm looking for the exact normal form, not the one up to $O(|x^5|)$ Those are two analogous problems, the first one of which I have already accounted for. Find ...