zeraoulia rafik
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How do I evaluate $\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\frac{1}{\sqrt{k}}$?
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14 votes

Let $a_n = 1 + \dfrac {1}{\sqrt{2}} + \cdots + \dfrac {1}{\sqrt{n}}$ and $b_n = \sqrt{n}$ for all $n\geq 1$. We have $$\lim_{n\to \infty} \dfrac {a_{n+1} - a_n} {b_{n+1} - b_n} = \lim_{n\to \infty} \...

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Prove that:$\sum_{i=1}^n\frac{1}{x_i}-\sum_{i<j}\frac{1}{x_i+x_j}+\sum_{i<j<k}\frac{1}{x_i+x_j+x_k}-\cdots+(-1)^{n-1}\frac{1}{x_1+\ldots+x_n}>0.$?
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13 votes

for any $0 \le x \le 1$, we have $$1 - \prod_{i=1}^n (1 - x^{a_i}) \ge 0$$ and hence $$f(x) = \sum_{i=1}^n x^{a_i} - \sum_{i < j} x^{a_i + a_j} + \cdots + (-1)^{n-1} x^{a_1 + \cdots + a_n} \ge 0.$$ ...

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How do I evaluate this integral $I = \int_{0}^{2 \pi} \ln (\sin x +\sqrt{1+\sin^2 x}) dx$?
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8 votes

In the First : $\sinh x = \frac{e^x - e^{-x} }{2} \Longrightarrow \sinh^{-1}x = \ln(x+\sqrt{x^2+1}).$ to get this just solve the equation $y=\sinh x$ to get the above inverse function (notice that $...

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How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?
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5 votes

According to an answer by rae306 on Art of Problem Solving: Using integration by parts: $$\int_0^\infty \ln\left(1+\frac{a^2}{x^2}\right)\,dx\,\,\begin{bmatrix}u=\ln\left(1+\frac{a^2}{x^2}\right)&...

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How to evaluate $\int \frac{\mathrm dx}{1+\sin x−\cos x} $?
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5 votes

$$\int\frac {\mathrm{dx}}{1+\sin x -\cos x}= \int\frac {\mathrm{dx}}{2\sin^2 \frac{x}{2}+2\sin \frac{x}{2} \cos \frac{x}{2}}\\= \frac{1}{2}\int\frac {\mathrm{dx}}{\sin^2 \frac{x}{2}+\sin \frac{x}{2} \...

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How do I prove this nice inequality $x+3\sqrt[3]{xy^2}\geq4\sqrt{xy} $?
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5 votes

Hint :$x+3\sqrt[3]{xy^2}=x+\sqrt[3]{xy^2}+\sqrt[3]{xy^2}+\sqrt[3]{xy^2}\ge 4\sqrt[4]{x\cdot\sqrt[3]{xy^2}\cdot\sqrt[3]{xy^2}\cdot\sqrt[3]{xy^2}}=4\sqrt{xy}$

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How do I show that:if$p$ is prime $>5$ then $p^4-20p^2+19$ is always divisible by $180$.?
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4 votes

Let $p^4-20p^2+19 = (p^2-1)(p^2-19)$, and $180 = 6^2\cdot 5$. But $p\equiv \pm 1\pmod{6}$, so $p^2\equiv 1\pmod{6}$, thus $p^2-1 \equiv 1-1 = 0 \pmod{6}$ and $p^2-19 \equiv 1-19 = -18 \equiv 0 \pmod{6}...

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How do I evaluate this integral :$\int \frac{\sqrt{-x^2-x+2}}{x^2}dx$?
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3 votes

$\begin{gathered} \int {\frac{{\sqrt { - {x^2} - x + 2} }}{{{x^2}}}} \,dx \hfill \\ = - \frac{{\sqrt { - {x^2} - x + 2} }}{x} - \int {\frac{{\left( { - 2x - 1} \right)}}{{2\sqrt { - {x^2} - x + ...

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find the minimum value of this integral when $1>t>0$, $f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x = ?$
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3 votes

First of all, it is obvious that the minimum is when $1>t>0$, (since $e^{-x^2}<1$ on the interval of integration). \begin{align*}f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x &= \int\...

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How do i find all integers $y$ such that $y^3 = 3x^2+3x+7$, where $x$ is also an integer?
3 votes

Try to take mod $3 \rightarrow y^3 \equiv 1 \pmod {3} \rightarrow y \equiv 1 \pmod {3} \implies y^3 \equiv 1 \pmod {9}$ Therefore $3x^2 + 3x + 7 \equiv 1 \pmod {9} \rightarrow 3x^2 + 3x \equiv 3 \...

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How do i find this : $\int \frac{1}{(x+a) \sqrt{x+b}}\ dx$, where $a > b > 0$?
Accepted answer
3 votes

Use the substitution $u^2 = x + b$. Let $I = \int\frac{1}{(x+ a)\sqrt{x + b}}\,dx $ Let \begin{align*} &u^2 = x + b\implies x + a = u^2 + a - b\\ &2u\,du = dx \end{align*} Then \begin{align*...

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How do i evaluate this $\int^\frac{\pi}{2}_0 \frac{\sec^2x}{(\sec x+\tan x)^{n}}\,dx = ? ?$
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2 votes

Maybe this answer from AOPS is helpful to you: $I =\int_0 ^\frac{\pi}{2}\frac{\sec^{2}x}{(\sec x+\tan x)^{n}}\ dx = \int_0 ^\frac{\pi}{2}\frac{1}{(\sec x+\tan x)^{n}} \cdot \frac{1}{\cos^{2}x}\ dx$ $...

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How do i find a closed form expression for $\sum_{k=0}^n \frac{(x-1)^k}{k+1}$?
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2 votes

We begin with the finite geometric series $f_n(z) = \frac{z^{n+1} - 1}{z-1} = \sum_{k=0}^n z^k. $ Integration term-by-term with respect to $z$ then gives $ \int f_n(z) \, dz = \sum_{k=0}^n \frac{z^{k+...

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Newton Raphson Step Size
2 votes

Hinit: What you meant is called halley method such that the Algorithm can be written as follow :$$ x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}[1-\frac{f(x_{n})}{f'(x_{n})}\frac{f''(x_{n})}{2f'(x_{n})}...

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Finding $\int_{0}^{e^2}(\frac{1}{\log{x}}-(\frac{1}{\log x})^2).\mathrm{d}x$
Accepted answer
2 votes

look the integral that you meant is :$$\int_{e}^{e^2}(\frac{1}{\log{x}}-(\frac{1}{\log x})^2).\mathrm{d}x$$ Hint : your integral doesn't converge ,you have a singularity at x=1 and should be begin ...

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show that the function $\{x_n\}\mapsto \sum_{n=1}^\infty 2^{-n}x_n$ is continuous
1 votes

Hint :$\sum_{n=1}^\infty 2^{-n}$ is a power of geometric series is already convergent and $\{ x_n\} \mapsto x_k$ is continuous for every $k$ ( Uniform convergence). pleas look ...

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Using The Riemann Zeta Functional Equation
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1 votes

pleas look this page 4 may you get what you want (P. Bourgade and J.P. Keating Seminaire Poincare)

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How do I show $\lim_{x\to\infty}f(x) = \lim_{x\to\infty} f '(x)=0$ if $\lim_{x\to\infty}f '(x)^2 + f(x)^3 = 0$?
1 votes

Suppose that : $\lim_{x\to\infty}f '(x)^2 + f(x)^3 = 0.$ Show $\limsup_{x\to\infty} f(x) = 0:$ Showing $\,\limsup_{x\to\infty} f(x) \le 0$ is easy. If $\limsup_{x\to\infty} f(x) < 0,$ then $f(x) &...

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$x^x = y$. Given $y$, find $x$.
1 votes

Hint :look Lambert W function. solutions represented as: $$x=\frac{\ln(y)}{W(\ln y)}$$

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integration of $1/x$ a counterexample to the rule
0 votes

Hint :if you seek for a proof just to look the following answer by "Mark Bennet" for related question Answer: Suppose we want to find the antiderivative of $\cfrac 1x$ - it doesn't come in the ...

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How do i find $\tan(\theta)$ such that : $\frac{16}{\sin^6(\theta)} + \frac{81}{\cos^6(\theta)}=625$??
0 votes

I cannot say that this is an elegant solution but oh well, i will use $s$ for $sin$, $c$ for $cos$ and $t$ for tan we know that $\frac{s}{c} = t$ so $\frac{16}{c^{6}*t^{6}} + \frac{81}{c^{6}} = 625$ ...

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How do i solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x$?
-1 votes

To solve: $3\sin^{3}x+2\cos^{3}x=2\sin x+\cos x$. Use the substitution $\sin x = \frac{2\tan \frac{x}{2}}{1+\tan^{2}\frac{x}{2}}=\frac{2t}{1+t^{2}}$ and $\cos x =\frac{1-t^{2}}{1+t^{2}}$. We become $t^...

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