# All Questions

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### Defining second value of the interval for bisection method

To find the interest rate of a loan (I'm talking about non linear formula that calculates Pmt) by using bisection method, I set the first value of the interval a = 10 ^ (-8), but I don't know how to ...
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### Aren't all digits of the $2^n$ for $n \ge 12$ even?

Define $a(n)$ is $2^n$ . All of digits of $a(11)$ (= 2048) is even. However, all of digits of $a(12)$ (= 4096) isn't even and all of digits of $a(13)$ (= 8192) isn't even. Aren't all digits of the ...
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### How to prove this statement on groups?

For a positive integer $n$. let $Φ(n)$ denote the number of elements $r∈Z$n such that $gcd(r,n)=1$. Show $Φ(mn)=Φ(m)Φ(n)$ for all $m, n∈N$ such that $gcd(m,n)=1$. The only thing I can come up with ...
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### Expressing the Christoffel symbols of the first and the second kind in terms of the metric tensor $g$ and expressing the symmetries

I am trying to expressing the Christoffel symbols of the first and the second kind in terms of the metric tensor $g=g_{ij}$ and express the symmetries of each with respect to the permutation of the ...
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### reference request: $C^k(\overline\Omega)$ as restriction of $C^{k}$ functions on $\Omega$

Let $\Omega\subset\mathbb{R}^d$ be an open set. $C^k(\Omega)$ is defined as the space of functions $f:\Omega\to\mathbb{R}$ such that $\partial^nf$ is continuous for $0\leq|n|\leq k$. There are ...
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### Deduce that: $\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}[mv\frac{du}{dx}+nu\frac{dv}{dx}]$

Deduce that: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv\frac{du}{dx}+nu\frac{dv}{dx})$$ When I differentiate $\frac{d}{dx}(u^{m}v^{n})$ I get: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv+nu)$$ Is ...
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### Solve the Following differential equation: $y''=2yy'$

Solve the following equation: $y''=2yy'$ My attempt: $y''=2yy'$ integrate on both sides: $y'=y^2y = y^3$ We therefore get: $y=\dfrac{1}{4}y^4$ But when I verify my answer in the original ...
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### If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to:

If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to: I do not know what to try to find the solution. A hint along with the explanation ...
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### Closed complex integral in an annulus

I have the function $$f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)},$$ and I want to find $$\oint_{|z|=1}f(z)dz.$$ What I know:  Let $A=\{z\in\mathbb{C}|r<|z|<R\}$ be the annulus with ...
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### How to express two variables in two other variables

If: $A=R\cos x$ and $B=R\sin x$ Then how can I express $R$ and $x$ in terms of $A$ and $B$ in a rigorous way? Meaning that I take the domain and range in account? I tried: $$\cos x=\frac{A}{R}$$ ...
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### f left continuous & strictly increasing; B Borel $\implies$ f(B) Borel?

How's it going? In an attempt to use the Radon-Nykodym theorem to bulldoze through the admission of measures by bounded variation & monotonic functions (sidestepping all that Caratheodory ...
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### How to calculate $\lim_{n \rightarrow \infty} n(1- \sqrt[n]{n}) = \infty$

Problem Show that $\lim_{n \rightarrow \infty} n(1- \sqrt[n]{n}) = \infty$ by the following consequence. Consider the sequence $(s_n)$ defined by $s_n=1 + 1/2 + \cdots + 1/n$. ...
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### Fraction with power denominator

I'm very confused as to how you're supposed to solve an inequality in which there is a fraction with a power as a denominator. Example: $$2^x + 8/{2^x} > 6$$ Thank you in advance!
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### Proper map and sequences in metric spaces

Let $f:X\to Y$ be a continuous map between metric spaces. Show that $f$ is proper if the following condition holds: For every sequence $x_n\in X$ such that $f(x_n)$ is bounded, $x_n$ is also ...
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### Evaluating a complex integral with a pole

I am asked to evaluate the integral $$\int_\gamma \frac{e^{2z^2}}{z^{77}}\,{\rm d}z$$ where $\gamma$ is a circle centre $0$ traversed once anti-clockwise. Clearly the integrand has a pole of ...
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### Hessian and Laplacian

Let $(M^n,g)$ be a smooth Riemannian manifold with a smooth boundary boundary $\partial M$ such that $Ric^M≥0$, and the second fund. form of $\partial M$ is $II\geq c>0$. Suppose that ...
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### Calculate a volume using a triple integral

I'm tring to find this volume using a triple integral in the form $dy$ $dx$ $dz$ However I think I'm evaluating the wrong integral because the result is 1 when the volume should be 1/6... Can someone ...
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### How much would this child get in the 30 days of the month of June.

This is the 10th question in my assignment. I'm not too sure if I got the common difference for this question correct. Here's the question: A child tries to negotiate a new deal for her pocket money ...