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Apr
24
comment Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?
@DanielV: No, I'm not reordering any terms, just inserting zeroes and using linearity. Explicitly, $0=(1 + \frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots)-(0+1+0+\frac{1}{2}+0+\frac{1}{3}+ \ldots)=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots = \log 2$.
Apr
23
revised Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?
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Apr
22
answered What is the expected number of suits in a hand of 4 cards?
Apr
22
comment Expected number of turns for SPROUT
My kids, in Oregon, play this game too :). They call it "Sproutball".
Apr
22
revised Expected number of turns for SPROUT
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Apr
22
answered Expected number of turns for SPROUT
Apr
21
comment Is there any interpretation to the imaginary component obtained when computing the geometric mean of a series of negative returns?
You should be taking the arithmetic mean of the logs of the price ratios, or the geometric mean of the actual price ratios ($p_{t+1}/p_t$), to determine the annualized (average) return. I don't think the geometric mean of the log-ratios is meaningful (or, indeed, actually used to compute returns in finance).
Apr
19
comment At time n, randomly choose a natural number ≤n. How long is it until a single number is chosen three times?
A simple way to write the problem is the following: "At time $n$, randomly choose a natural number $\le n$. How long is it until a single number is chosen three times?"
Apr
18
awarded  Enlightened
Apr
18
awarded  Nice Answer
Apr
17
awarded  Revival
Apr
16
answered $\lim_{n \to \infty} \frac{1}{2n}\log\binom{2n}{n}$
Apr
16
comment How is this commutator property $[a,b]=bx$ called?
Don't know. Seems like it arises for exponentials of operators, for instance. E.g., $[\partial_x, e^{kx}] = ke^{kx}$.
Apr
16
answered A number $n$ which is the sum of all numbers $k$ with $\sigma(k)=n$?
Apr
16
comment How is this commutator property $[a,b]=bx$ called?
If $x$ is proportional to $a$, then this commutation relation defines the quantum plane: $ab = qba$, or $[a,b]=ab-ba=(q-1)ba$. I don't know if it has a special name in general.
Apr
15
comment Chess Probability 8 rooks
It matters where the blocked square is, I think. In the center?
Apr
13
comment Writing numbers as a sum of 2s and 3s
This "reaching back one step further" Fibonacci sequence is also called a "skiponacci sequence"; with these initial conditions ($c_0=1$, $c_1=0$, and $c_2=1$), you get a third, linearly independent solution with respect to two others that are known and well-studied: the Perrin sequence and the Padovan sequence. The growth rate is the so-called "plastic constant".
Apr
13
answered A big “smallest” number
Apr
13
revised A big “smallest” number
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Apr
13
comment combinatorics - pigeonhole principle - 2
There are $2^n-1$ different non-empty subsets of $X$. If $2^n-1 > n^2$ (which is true for $n \ge 5$), then two different non-empty subsets of $X$ must have the same sum modulo $n^2$. Assign $+1$ to elements in only the first subset and $-1$ to elements in only the second subset.