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1h
awarded  elementary-number-theory
1h
comment Is my argument correct?
Two reasons why 1000 can't be $3^m$: (1) prime factors; (2) 3^6 = 729 < 1000 < 3^7 = 2187$.
1h
answered Which are Linear homogeneous recurrence relations
3h
answered Calculate $\lim_{x \to 0} \frac{e^{3x} - 1}{e^{4x} - 1}$
4h
answered System of linear equations: get approximate solution with non-negative coefficients
4h
answered What are statements about the natural numbers where induction is impossible or unnecessary to prove?
4h
comment Examples of apparent patterns that eventually fail
My humor is 71% dark. It may not be the best tasting, but it is better for you.
5h
comment How can I determine the value of $a_1 + \displaystyle\sum_{i = 1}^{2012}\frac{a_{i + 1}^3}{a_i^2 + a_ia_{i + 1} + a_{i + 1}^2}$
Sometimes I am right.
5h
comment Are logarithms radicals?
Would it help to know that $\log(x)$ grows more slowly than $x^c$ for any $c > 0$?
5h
comment transforming an equation into a difference equation
What are the $s_i$ and $a(x)$?
5h
answered Examples of apparent patterns that eventually fail
5h
comment How can I determine the value of $a_1 + \displaystyle\sum_{i = 1}^{2012}\frac{a_{i + 1}^3}{a_i^2 + a_ia_{i + 1} + a_{i + 1}^2}$
Don't know where it's from, but this is the type of problem that is readily solvable using some unexpected insight and almost totally impossible otherwise.
5h
comment Show that the following sequence converges for $ 0 < a < e $ and diverges for $ a \ge e$
By Stirling's theorem, $n! \approx c\sqrt{n}(n/e)^n$, so $e^n n!/n^n \approx c \sqrt{n}$ so the sum diverges. This can be proved by reasonably elementary means using the properties of log and exp.
5h
comment How can I determine the value of $a_1 + \displaystyle\sum_{i = 1}^{2012}\frac{a_{i + 1}^3}{a_i^2 + a_ia_{i + 1} + a_{i + 1}^2}$
Yep. That's a contest problem, alright.
5h
comment If I flip $n$ coins, what is the probability of at least $3/4$ of them coming up heads?
In other words, $p(n/4) \approx \sqrt{c/n} d^n$ where $c = 8/(3\pi)$ and $d = 2/(3^{3/4}) \approx 0.877$.
5h
comment If I flip $n$ coins, what is the probability of at least $3/4$ of them coming up heads?
If you use Stirling, you get my approximation.
5h
comment If I flip $n$ coins, what is the probability of at least $3/4$ of them coming up heads?
btw, if you want to estimate the probability of getting exactly 1/4 heads, the answer is here: math.stackexchange.com/questions/1208016/… It turns out that $\binom{4n}{n} \approx \sqrt{\dfrac{2}{3\pi n}}\left(\dfrac{4^4} {3^3}\right)^{n}$ so that $\frac1{2^{4n}}\binom{4n}{n} \approx \sqrt{\dfrac{2}{3\pi n}}\left(\dfrac{2^4} {3^3}\right)^{n}$.
5h
comment If I flip $n$ coins, what is the probability of at least $3/4$ of them coming up heads?
You have to use the Normal approximation.
21h
comment Visualize $z+\frac{1}{z} \ge 2$
Since the geometric mean of z and 1/z is 1, erect a unit line prependicular to the end of a segment of length z. Connect the other end of the segment to the end of the unit segment, and draw the perpendicular to it. That will intersect the extended segment of length z in a segment of length 1/z.
1d
answered What is an adjective for “weaker than weak”?