# All Questions

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### Help with the proof that the sum of all the roots of a complex number is zero

If a complex number $z \neq 0$ has n roots, then each root can be expressed as: $$z_j=(\sqrt[n]{r}) e^{ \left( {i \theta+2\pi j }/{n} \right) }$$ For $j=0,1,2,...,n-1$ Thus, the summation of all ...
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### Explanation of definition of normal subgroup

Recently I am studying group theory to know about Orbit-Stabilizer Theorem and I'm a novice learner.I got a definition of normal subgroup as mentioned in the link: ...
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### Time taken to give answer if probability is given.

This is a question that I am struggling with: Since the password is periodically changed, you would like to know the answer as soon as possible. So you decide to interrogate the minions in an order ...
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### How can I prove or disprove that every series converges to philosophy? [on hold]

Let $x_0$ be any wikipedia page. Let $x_n$ be the page that the first non-etymological link on $x_{n-1}$ leads to. How can I prove that $x_n$ eventually converges to philosophy? Technically it seems ...
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### Use of Laplace transform to solve initial value problem.

--Short Explanation: I have to say I am going crazy with this problem as it does not give me the same as the suggested solution in the book: Problem: $y''-7y'+10y=9\cos{t}+7\sin{t}$ $y(0)=5$, ...
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### Why is the closedness of the set on which $f = g$ immediate, when proving the Identity Theorem?

It's a short proof given on Wikipedia, and I understand the argument for why the set on which $f = g$ must be open, but I'm not sure why closedness of the set is obvious. Apparently, this comes from ...
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### How do you convert different bases?

I know how to convert any number into base 10 by using the below method. Write (6712)base 8 in base 10. Ans: $6 \times 8^3 + 7 \times 8^2 + 1 \times 8^1 + 2 \times 8^0 = 3530_{10}$ However, I am ...
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### Complex integrate and residues

Evaluate the integral of that $f(z)=\frac{z+1}{z^2-2z}$ around the circle $|z|=3$ oriented counterclockwise First I found that singularity points are $z=0,z=2$ ...
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### Orthogonality on a circle

These theory questions always throw me for a loop. Ok, the question is: Suppose that AB is the diameter of the circle with center O and that C is a point on one of the two arcs joining A and B. Show ...
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### if $\lim a_n = \infty$ and $\lim b_n = B$, then $\lim (a_n+b_n) = \infty$

I'm having trouble starting the proof not sure exactly how to go about it. So far I know that for a sequence to go to infinity it means that for all $n >0$ there exists $n_0$ for all $n$ greater ...
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### A function with midpoint-linear derivative is a quadratic polynomial

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $$f'\left(\frac{a+b}{2}\right) = \frac{f'(a)+f'(b)}2,\quad \forall a,b\in\mathbb{R}$$ Prove that $f$ is a polynomial of ...
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### Probability of Independent Events individual vs in series

I understand that independent events (such as a fair coin flip) should not be viewed in succession. For example, if you flip heads 10 times in a row, the odds of flipping the next coin heads is still ...
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### Next book in learning Differential Geometry

I have just finished the book "Manfredo P. do Carmo - Differential Geometry of Curves and Surfaces". My aim is to reach to graduate level to do research, but articles are not only too advanced to ...
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### Understanding a chart about likelihood

I am learning stats from All of Statistics 1e. I understand that likelihood is the probability of a parameter, given some data. So if you flip an unknown coin 1000 times and get 500 heads a parameter ...