Adrian Ratnapala
  • Member for 8 years, 11 months
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2 answers
5 votes
251 views
How can this result in Thermodynamics be rigorously proved?
4 votes

I know this is not really what the OP wanted, but to me $W = \int {\rm d}{\bf r \cdot F}$ is a bad starting point. It's a bit like a schoolboy saying Pythagoras' theorem is $a^2 + b^2 = c^2$ and ...

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4 answers
1 votes
2k views
Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ is convergent in $\mathbb{R}$
2 votes

Let $\phi$ be the golden ratio, which is the only positive real such that $\phi = 1 + 1/\phi$. Now consider the sequence $\epsilon_n = a_n - \phi$. We seek to prove the invariant: $$|\epsilon_n/\...

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2 answers
0 votes
196 views
Does this integral make sense in some way?
Accepted answer
2 votes

In most fields the integrand needs a $\mathrm{d(something)}$ for it to make sense. This is what makes it in a hand-wavy way "infinitesimal". However, in some fields of maths you are allowed to say ...

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3 answers
1 votes
87 views
Prove that *BIG'* = *BIG* - *Little* (set difference) is uncountable.
1 votes

If $Little$ is countable, there is a 1-1 mapping between it and the integers, indeed between it and just the odd integers. Now suppose $BIG'$ is countable then must be a mapping between it and the ...

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1 answers
0 votes
297 views
Leaky integrator in novice terms
Accepted answer
1 votes

An example might be a resistor-capacitor filter in electronics. The charge on the capacitor will be the integral of the current you pour into it (and the voltage across it proportional to the charge)....

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