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Mar
7
comment “What if” math joke: the derivative of $\ln(x)^e$
This has nothing to do with xkcd, besides being by the same author.
Mar
3
awarded  Great Answer
Feb
6
awarded  Nice Answer
Feb
2
awarded  Good Answer
Feb
2
awarded  Enlightened
Feb
2
awarded  Nice Answer
Jan
23
awarded  Good Answer
Jan
23
comment Curious Binomial Coefficient Identity
Just for completeness, on deriving $B(x)$: note that $\sum_{n=0}^{\infty}\binom{n}{k}x^n = \sum_{n=k}^{\infty}\binom{n}{k}x^n = x^k\sum_{n=0}^{\infty}\binom{n+k}{k}x^n =x^k\sum_{n=0}^{\infty}(-1)^n\binom{-k-1}{n}x^n=x^k(1-x)^{-k-1}$ as $$\binom{-k-1}{n} = \frac{(-k-1)(-k-2)\cdots(-k-n)}{n!}=(-1)^n\frac{(n+k)\cdots(k+1)}{n!} =(-1)^n\binom{n+k}{k}.$$
Jan
22
answered Curious Binomial Coefficient Identity
Jan
22
comment Curious Binomial Coefficient Identity
@anorton: Quite clearly from context, $a_n = \binom{n}{k}$ (for some/any fixed $k$).
Jan
3
awarded  Nice Answer
Dec
8
awarded  Caucus
Dec
7
comment Can I get a decimal number does not contain a similar consecutive double-digit???
Try $12/99 = 0.12121212...$
Dec
1
awarded  Nice Answer
Nov
29
comment Prime numbers stretch to infinity, but what about the distance between them?
But Zhang's result does prove the "probably" at the top: it proves that $\liminf_{n \to \infty} (p_{n+1} - p_n)$ is finite, and therefore the $\limsup$ and $\liminf$ are different, i.e. the limit definitely does not exist.
Oct
2
awarded  Enlightened
Oct
2
awarded  Nice Answer
Sep
30
awarded  Explainer
Sep
24
awarded  Autobiographer
Sep
23
comment Does the number pi have any significance besides being the ratio of a circle's diameter to its circumference?
@Ant: My comment was a reply to asmeurer's speculation that it had to do with "angles and the 2D lattice" -- my comment was not a reply to anything in this (damiano's) answer.