# All Questions

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### logic: two simple math contradictions

1.The contradiction of the sentence: - There is a greater number than a million. can be stated as follows: - There is a number which is not greater than a million. 2.and the contradiction of the ...
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### Calculus Exponential Function Help

I need help finding the limit of this expression. I do not want the answer, I want to learn the steps to approach this problem. $$\lim_{x \to\infty} 1.082^x$$ I know if $a > 1$, the limit is ...
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### Series Sum up to N terms

I have been trying to find the sum of a series given by $t(n) = \frac{1}{2^n-1}$, up to N terms. All I could do is to see that the difference of the successive denominators form a GP. Kindly help me ...
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### Showing the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$

Problem: If $\sum a_n z^n$ and $\sum b_n z^n$ have radii of convergence $R_1$ and $R_2$, show that the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$. Is the following proof ...
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### Integration by parts of the Logarithmic Integral

I am doing work on analytic number theory, and I am currently looking at the Prime Number Theorem, that is $$\pi(x) \sim Li(x)$$ Some of my sources say that I can do integration by parts on the ...
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### Intuition. Cauchy criteron for Riemann integrability (Spivak pp 239, S. Abbott pp 189 thm 7.2.8)

1. Why $\inf U(f,P') \le U(f, P)$ and $\sup L(f, P') \ge L(f,P)$? I tried to research but I can't find where Spivak defined it $P'$? 2. Why are there two partitions P', P''? Not the same? ...
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### Approximation to $\sqrt{\cos(\theta)}$?

I have this formula, (it is just the law of cosines angle formula): $$d = \sqrt{a^2 + b^2 - 2ab \ cos(\theta)}$$ Here is my issue. I am wondering if there is a way to 'extract' the $cos$ term. My ...
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### Rules of inferences problem

I have a exercise that I have to prove the validity of the following arguments using rules of inferences But there is a exercise I have no idea how to prove it at all. Problem shows as below. P-> ...
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### How to prove this Logarithmic identity?

I am just learning logs, and can't get this one out ? How to prove that $$\log_a (b) \cdot \log_b (c) = \log_a (c)$$ in the format : log [base]([argument]) ...
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### Free fall with resistance: solution to the ODE

I'm having trouble solving this ODE: $$\ddot x = \mu \dot x^2 - g, \space \space x(0)=x_0$$ This is the ODE that determines the equation of motion of an object with air resistance. $\mu$ is a ...
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### Prove that there exist $c_n>0$ such that $f^{(n)}(c_n)=0 \forall n$

Let $f \in C^\infty([0,+\infty),\mathbb{R})$, and $f(0)=\lim\limits_{x \to \infty}f(x)$ Prove that there exist $c_n>0$ such that $f^{(n)}(c_n)=0$ for all $n$ integer, where $f^{(n)}$ is ...
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### Who proved Fundamental Theorem of algebra using Liouville's theorem?

One of the most famous proofs of the Fundamental Theorem of Algebra involves Liouville's theorom stating that a bounded entire function in constant. Who first came up to the idea of deriving FToA ...
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### Equivalence relation and quotient set

I'm studying for a test and got stuck in one question regarding equivalence relations and quotient set. Here is the question: Let $F=\mathbb{R}\to \mathbb{R}$ be the set of functions from ...
Conjecture The interval of the natural number line $[ap_{n}, (a+1)p_{n}]$ contains a member $e$ that is not divisible by any prime number $p_{m}$ less than or equal to $p_{n}$, if $(a+1) \leq ... 0answers 225 views ### Condition for a block matrix to be positive semi-definite Let's say we have two positive semi-definite matrices$A$and$B$and a negative real$\lambda$. What would be the conditions for the matrix$M$, defined as follows, to be positive semi-definite too ? ... 1answer 101 views ### Proof of Extreme Value Theorem. Modus operandi. (S. Abbott pp 115 t4.4.3) By pp 115 Abbott Theorem 4.4.2, we know that$f([a, b])$is bounded above and below.$f([a, b])$contains [a,b] ergo it is clearly nonempty). By the agency of the Axiom of Completeness, it has a ... 0answers 64 views ### Show using Stirling's approx. that$\log\binom{n}{\gamma n} = nh(\gamma) -\frac{1}{2} \log n + O(1).$Let$h(p) = -p \log p-(1-p)\log (1-p)$denote the binary entropy of a Bernoulli distribution when the probability of observing a zero is$p$, where$\log$denotes the logarithm to base 2. Show, using ... 2answers 43 views ### Limit of$x^ny^n$when$n \to \infty$Please, someone could help me in solving the following limit. Let$x \in \mathbb{R}$,$y \in \mathbb{R}$and$n \in \mathbb{N}$. Also consider that$0<x<1$and$1<y<\infty$. $$\lim_{n ... 1answer 54 views ### Prove that f^{-1} is continuous Let M be a compact metric space (N is a metric space too) and let f:M \to N be a continuous bijection. Prove that f^{-1} is continuous. My proof. Let A \subset M be closed. Then A is ... 4answers 129 views ### Probability & Statistics: Random variables I have a problem similar to the well-known "Coupon Collector Problem." A box of a certain brand of cereal comes with a special toy. There are 10 different toys in all. How many packs you will need ... 0answers 63 views ### Notation of List other than Set, and related operations? we all know that a single capital letter (e.g., S) usually represent a set (containing non-duplicate objects) and we can write a number of operations on set such as |S|, |S| union |S'| etc. ... 1answer 70 views ### How to deal with the following problem of correlated random variables? I have the following information: \left[ \begin{array}{l} {X_1}\\ \vdots \\ {X_K} \end{array} \right] are correlated random variables with (zero mean, unit variance) covariance matrix \left( ... 0answers 62 views ### Induced measure on dual group and kernel of fourier transform Let G be a locally compact Hausdorff abelian group (LCA). Then the Fourier transform gives a map from functions on G to functions on \hat G$$f\mapsto \hat f(\xi) = \int_G f(x)\Psi(x,\xi) ... 0answers 317 views ### A Ramanujan-like summation: is it correct? Is it extensible? I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. Looking at the series $$s(1/2,2) = (1/2)+(1/2)^4+(1/2)^9+(1/2)^{16}+...$$ and more general $$s(b,p) ... 1answer 189 views ### Does there exist a Lebesgue nonmeasurable set whose measurable subsets are all null sets? Let \mu be the Lebesgue measure. Let A be a Lebesgue-nonmeasurable set. Define S:=\{E\subset A : E\text{ is Lebesgue measurable}\}. Does there exists a nonmeasurable set A satisfying ... 1answer 20 views ### Confusion with notation P[event in dy] What does it mean when people write: \Pr \{ B \in dy \}\ What I don't understand is what dy in there means. 2answers 62 views ### Logarithmic Differentiation When do we use : \ln(ab) = \ln a + \ln b and when do we use : \ln |y| = \ln |f_1(x)| + \ln |f_2(x)| + \cdots + \ln |f_n(x)| ? It is stated that we use the second form of log differentiation ... 0answers 300 views ### General (Set Builder) definition for a relation composed with itself n times Questions What does the set builder notation for S\circ R look like? I'm having the most trouble knowing when there is too much information or not enough information on either side of the 'such ... 2answers 57 views ### need to show a complex function is continuous on C (complex plane) Prove$$f(z) = \sum\limits_{n=0}^\infty \frac{z^{2n}(-1)^{n}}{(2n)!}$$is continuous everywhere on \mathbb{C} I want to show, for each \epsilon > 0, there is some \delta > 0 such that ... 2answers 24 views ### Compute limit of sequence Let (x_n) be real sequences such that x_{1}=\dfrac{1}{3}, x_{2n}=\dfrac{1}{3}x_{2n-1}, x_{2n+1}=\dfrac{1}{3}+x_{2n}, n=1,2,\cdots . Compute$$\lim_{x \to \infty} \sup x_{n} \text{ and } \lim_{x ... 2answers 90 views ### Open cover of$(0,1)$Suppose the open interval$(0,1)$is given absolute value topology. Form$\{U_n \}$,$n=1,2,3…$, where$U_n=(\frac{1}{n+1},1)$. Prove that$\{U_n \}$is an open cover of$(0,1)$for all$n∈N$. Show ... 2answers 47 views ### Calculating$e^{3x} \pmod{27}$I am following some notes that say that$\exp(3x) = 1 + 3x + \frac{9}{2}x^2 + \frac{9}{2}x^3 = 1 +3x+18x^2+18x^3 \pmod{27}$I can't understand this. Firstly why do we stop the expansion after 4 ... 2answers 222 views ### Initial Segments and Initial Sections of Posets For a set A with a partially ordering <=, define the following 1) A subset s(x) of A = {y in A such that y <=x} 2) A subset S of A with the property that for every x in S then all y in A ... 0answers 164 views ### Infinitesimal thickening of a smooth closed subscheme Let$A$be a noetherian ring (if it is useful I can assume that$A$is an algebra of essentially finite type over a field) and$I \subset A$is an ideal s.t.$A/I$is smooth. Is it true that extension ... 2answers 120 views ### How find this sum$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)(n+3)\cdots(n+2014)}\$ [duplicate]
Find the sum $$\sum_{n=1}^{\infty}\dfrac{1}{n(n+1)(n+2)(n+3)\cdots(n+2014)}$$ My idea: since ...