# Tagged Questions

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### Evaluating $\lim_{n\to\infty}\left(\frac{1-i}{4}\right)^n$

It's been a while since I have taken Calculus II so my experiences on sequences and series has gone down the drain. I'm trying to find the limit of the sequence ...
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### A conditional probabilty question.

Question: $8$ identical balls are randomly distributed into $8$ boxes. Given first box and second box are not both empty, find the probability that first box is not empty? $A:=$ B1 is not ...
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### Two probability questions.

I have two questions. (1) solution(1): Sample size $=|S|=12^{20}$ $11^{20}\rightarrow$ guarantee that one box is empty. $10^{20}\rightarrow$ guarantee that two boxes are empty. ...
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### how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty

Question: how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty. I understand case-1. But I cannot understand a part of answer ...
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### Pigeonhole Principle and maximum length of the repeating section

The question I have is, when 5 / 20483 is written as a decimal, what is the maximum length of the repeating section of the representation? I believe I need to divide 5 by 20483 which is equal to ...
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### Explain why the description defines a Boolean algebra.

Here is the exercise I am trying to figure out. Let A = {a,b} and list the four elements of the power set P(A). We consider the operations + to be $\cup$, . to be $\cap$, and complement to be set ...
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### countably compact subset of a metric space is sequentially compact?

I'm currently trying to get my head around a proof in Lipschutz [1]. Solved exercise 21 of Chapter 11 (page 164) asks for a proof of the following statement: Let $A$ be a countably compact subset of ...
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### counting and probability question - help needed

I am stuck on how to start this exercise. Any help is welcome. An instructor gives an exam with 14 questions. Students are allowed to choose any 10 to answer. Suppose the exam instructions specify ...
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### How to write Pfaffian differential equation

I am studying the example. And I dont understand how to write the pfaffian diff equation in the first line. What is its formula?
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### Show that the curves are circle.

For example, this question is also similar to the previous question I have asked, which is the following link; How to show the curves are conics. Question: Solve the equation ...
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### How to show the curves are conics.

Solve the equation $$\frac{dx}{x^2+b^2} =\frac{dy}{xy-bz}=\frac{dz}{xz+by}$$ How to show the curves are conics. First of all , I need to find its integral curves. I tried to solve. But I get ...
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### How to find required surface.

Hi I am studying this question. I understand all parts expect for the part I posted below. Please explain it. Thank you so much. I am new learner of PDE.
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### Solving Lagrange equation.

Please help me solving the equation. I found one of its solutions: $c_1=(xz)/y$ But another one is given as $c_2=(x^3/y)+x$ in the text book. But I cont find. Thank you.
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### find the general integrals of the given P.D.E

I tried to find the general integrals of the given P.D.E in the yellow box. And I found $c_1$. But I cannot find another one, say $c_2$. Please help me finding $c_2$. Thank you.
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### What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$?

What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$? I know its answer: 0, -2, -1+i, -1-i. But I dont know how to find? Please show me this. I know this is so trivial, but important ...
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### Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist.

Let $R$ be the domain obtained by removing the non negative real numbers from $\Bbb C$. Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist. ...
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### Classify all invertible meromorphic functions

I am studying complex analysis. And I see a topic, namely meremorphic functions But I cannot find any enough information about this function. So I have insufficient knowledge about this topic. ...
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### Conformal map example $f(z)=e^z$

I an studying the example-1. I understand $f(z)=e^z$ has a nonzero derivative at all points, hence it is everywhere conformal and locally $1-1$. But I dont understand th part I underlined with ...
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### Trying to understand an exercise using factorials with induction

Exercise: Prove that (n + 1)! - n! = n(n!) for any n $\ge$ 1 Given Answer: I will skip the basic step since I understand that part. (n + 2)! - (n + 1)! = (n + 1)!(n + 2) - n!(n + 1) I understand ...
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### If $\gamma$ is spherical, the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Please help me doing this question. In fact, there is its solution as I posted below. But I don't understand the answer. Please explain this more clearly. Thank you for helping. ...
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### How to obtain $y$

The question was written with dark-blue pen. And I tried to solve this question. I obtained $x$ as it is below. But I cannot obtain $y$ Please show me how to do this. By the way, $\gamma (t)$ ...
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### how to find the signed normal

$$\gamma (t)= (R\cos (t/R), R\sin (t/R))$$ $$\dot {\gamma (t)}=(-\sin (t/R), \cos (t/R))$$ $$n_s= (-\cos (t/R), -\sin (t/R))$$ where $n_s$ is the signed normal. the instructor has found the $n_s$. ...
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### calculation of vector product $\ddot{\gamma} (t) \times \dot{\gamma} (t)$

$$\ddot{\gamma} (t) \times \dot{\gamma} (t)=(-a\cos t, -a\sin t, 0)\times (-a\sin t, a\cos t, b)$$ The writer will get the following result $$(-ab\sin t, ab\cos t, -a^2)$$ but I don't know how to ...
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### how to calculate the curvature of an ellipse

how can I compute the curvative of an ellipse given by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ do i need to take $x=acos(t)$ and $y=bsin(t)$? please show me a way how to solve this? thank you for ...
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### solve this equation $z(z+y)dx+z(z+x)dy=0$

I need to solve this following equation $$z(z+y)dx+z(z+x)dy=0$$ I get this from above equation $$\frac{dx}{z(z+x)}+\frac{dy}{z(z+y)}=0$$ After there, I dont know what I need to do.
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### Find solution(primitive) of the equation

I want to find its solution of the following equation $$ydx+xdy+2zdz=0$$ answer: Keeping $z$ constant; I obtain that $$ydx+xdy=0$$ or $$\frac{dx}{x}+\frac{dy}{y}=0$$ Then I get $$U(x,y,z)=xy$$ ...
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### How to obtain the last ratio $\frac{d(x+z)}{(x+z)}$

I am studying example-2.3 In the first line, it says that "the numerators and denominators in the first and last ratio" And the following is obtained $$\frac{d(x+z)}{x+z}=\frac{dy}{y}$$ But I ...
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### The Decision of three methods of the solutions $dx/P=dy/Q=dz/R$

Question: (A) $$\frac{adx}{(b-c)yz}= \frac{bdy}{(c-a)xz}=\frac{cdz}{(a-b)xy}$$ (B) $$\frac{dx}{xz-y}=\frac{dy}{yz-x}=\frac{dz}{1-z^2}$$ These are simultaneous diff eq. of the first order and the ...
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### Calculation of sum by using residue theory

I am studying an example about the calculation of a summation by using residue theory. I understand how to calculate the sum in general -the frame of the solution way-, but I dont know how some parts ...
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