Questions tagged [summation]

11,422 questions
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Convergence of a double series $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{(m+n)^2}$ [duplicate]

Does the series $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{(m+n)^2}$ converge?
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Inequality. $\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}+\frac{c}{\sqrt{a}}\geq3$

Let $a,b,c \in (0, \infty)$, with $a+b+c=3$. How can I prove that: $$\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}+\frac{c}{\sqrt{a}}\geq3 ?$$. I try to use Cauchy-Schwarz rewriting the inequality like : ...
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$\sum_{k=1}^{100}k\cdot 2^k\geq 99\cdot 2^{101}$ solve geometrically?

I've succeeded proving it using induction, but I wondered if there exists a prettier solution, maybe prooving it geometrically with areas of rectangles or some other way.
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How to find $\lim_{n \to \infty}\Sigma_{i=1}^n\frac{\sin(\pi x)^2}{i^2}$? [closed]

I am trying to find: $$\lim_{n \to \infty}\sum_{i=1}^n\cfrac{\sin(\pi x)^2}{i^2}$$ using maple command: limit(Sum(sin(Pi*x)^2/i^2, i = 1 .. n), n = infinity); ...
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Generate random vibration with zero mean for the motion of a point in 3D space

I want to simulate a point that moves with random vibration around a mean position (let's say around the position $[X, Y, Z] = [0,0,0]$). The first solution that I found is to sum a couple of ...
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Calculate the maximum value of $\sum_{cyc}\frac{1}{\sqrt{a^2 + b^2}}$ where $a, b, c > 0$ and $abc = a + b + c + 2$.

$a$, $b$ and $c$ are positives such that $abc = a + b + c + 2$. Caculate the maximum value of $$\large \frac{1}{\sqrt{a^2 + b^2}} + \frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}}$$ This ...
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