# All Questions

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### Why does Newton's method approximate $\sqrt{z}$ to be $z - \frac{z^2 - x}{2z}$?

According to a document I am reading, I can approximate the square root of some real number $x$ via "Newton's method" by repeatedly taking $z = z - \frac{z^2 - x}{2z}$ after beginning with some random ...
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### Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
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### Compute $\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space$

How would you approach $$\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space?$$ The way I see here involves Dirichlet kernel. I wonder what else can we do, maybe some easy/elementary approaching ...
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### A differentiable injective function with Lipschitzian Inverse

I'm having difficulty with the following question which was given to me following studying the inverse mapping theorem. Let $U\subseteq\mathbb{R}^{n}$ be an open set and let $f:U\to\mathbb{R}^{n}$ ...
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### How can I systematically find the roots of $x^4 + 1?$ Is there some algorithm? [duplicate]

Possible Duplicate: How to find the root of $x^4 +1$ What algorithms can be used for finding all roots of the given polynomial: $$x^4 + 1 = 0$$
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### Cluster point of $a_{n}:=n+(-1)^{n}n$

I am trying to find the cluster point of the sequence $a_{n}:=n+(-1)^nn$. Can you please check my solution? The subsequence diverges for increasing even $n$ since $2n$ grows infinitely. The ...
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### Solutions of Diophantine equations in Natural numbers

The one of solution of $x^4 - 2y^2 = -1$ is $x = 1$ and $y = 1$. However, the solution $(1, 1)$ of $x^4 - 2y^2 = 1$ is failed. We know $x = 1$ and $y = 1$ is small integers and we can check by trail ...
I need help in proving the following popular claim A continuous time and stationary Markov jump process obeys the detailed balance equations $$P(x)q(x,x') = P(x')q(x',x)$$ where $q(\cdot,\cdot)$ is ...
### How can I algebraically prove that $2^n - 1$ is not always prime?
This question is from Elementary Number Theory by W. Edwin Clark. Is $2^n - 1$ always prime, or not? Prove. Is this a start? $x^n - 1 = ( x - 1)(1 + x + x^2 \cdots x^{n - 1})$. So, \$2^n - 1 = ...