# Questions tagged [alternative-proof]

If you already have a proof for some result but want to ask for a different proof (using different methods).

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### Is there an alternative way to solve this trivial problem?

I dealing with a trivial problem: Let $f(x) = xy$ be some area. Given $1500 = 15x + 6y$, find $x$ and $y$ so that the area is maximized. Usually pretty easy when you can just plug in $(1500 - 15x)/6$...
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### Is this proof circular or is it valid?

I am playing with the chernoff bounds and derived a bound from its additive version. The method I used seems a little suspicious and circular in logic, but I cant figure out what the mistake is. ...
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### Probability Proof(Finan #5.1.15)

Show that $P(A|B)>P(A)$ if and only if $P(A^c |B)<P(A^c)$. We assume that $0<P(A)<1$ and $0<P(B)<1$ ( Finan #5.1.5) Taking the complement of probabilities reverses the sign of the ...
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### Alternative Proof that ln(2) is Irrational

Is the following a correct proof that $\ln(2)$ is irrational without using the fact that $e$ is transcendental? The power series $\ln(1+x)=\sum_{k=1}^\infty \frac{(-1)^{k+1}x^k}{k}$ for $x=1$ gives ...
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### Simpler proof of identity $\sum_{n=1}^{\infty} \frac{H_n^2}{n2^n} = \frac{7}{8} \zeta(3)$

The identity in question can be obtained by first proving $$\sum_{n=1}^{\infty} \frac{H_n^2}{n} z^n = - \frac{1}{3} \log^3(1-z) -\log(1-z) \text{Li}_2(z) + \text{Li}_3(z), \hspace{0.5cm} |z|<1,$$ ...
1 vote
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### A simpler approach to Brezis' Proposition 8.5

I'm reading a result from Brezis' Functional Analysis Proposition 8.5. Let $u \in L^p(\mathbb{R})$ with $1<p<\infty$. The following properties are equivalent: (i) $u \in W^{1, p}(\mathbb{R})$, ...
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### Evaluating $\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{ij(i+j)^2}$

I was playing around with double sums and encountered this problem: Evaluate $$\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{ij(i+j)^2}$$ It looks so simple I thought it must have been seen before, ...
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### Is every finite scheme affine?

$\def\sF{\mathcal{F}} \def\sO{\mathcal{O}}$Today I wondered is every finite scheme affine? and I found a proof using Serre's criterion on affiness; however, I don't know if I may be using a ...
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### Find the Length of Parallel Line Through Trapezoid's Diagonals Intersection with given Bases Lengths

The problem: A straight line is drawn through the point of intersection of the diagonals of a trapezoid, which is parallel to the bases and intersects the sides of the trapezoid at points $M$ and $K$....
1 vote
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### Two different expressions from solving $s(n, m) = s(n-1, m) + s(n-1, m-1)$, alternate proof they are equal.

Before reading this please quickly look at this question I asked before, and the accepted answer. Upon further inspection, I found another solution to the recurrence relation, namely (the reciprocal ...
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### The fourth power of any integer has the form $8m$ or 8m + 1 for some integer m.

In my textbook I came across this problem: Prove that the fourth power of any integer has the form $8m$ or $8m + 1$ for some integer $m$. I know how to solve the algebra but don't understand the ...