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1d
comment Find the radius of convergence of $\sum_{n=1}^\infty n!(2x-1)^n$
You've fixed your error now... you need $\lim_{n\rightarrow\infty}(1/(n+1)) > |2x-1|$. Since the left-hand limit is zero, the inequality is never satisfied, and so the radius of convergence is zero.
1d
comment Two dice tossed 10 times. Probability of rolling same number knowing amount the number rolled?
You really just want the probability that the 4 times you hit with the first die lie among the 8 times you missed with the second die. This probability is ${{8}\choose{4}} / {{10}\choose{4}}=1/3$. (Equivalently, the probability that the 2 times you hit with the second die lie among the 6 times you missed with the first die is ${{6}\choose{2}} / {{10}\choose{2}} = 1/3$ as well.)
1d
answered Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$?
2d
answered Does an analytical form exist for the following integral
2d
comment Partitions of a number with greatest product
Why $3$? If you break it into $n$ parts, the product is maximized when the $n$ parts are about the same size, and is then about $(k/n)^n = \exp(n (\log k -\log n))$. This is maximized when the number of parts is $n = k/e$; i.e., each part has size $k/n = e \approx 3$.
Apr
16
comment Developing a Process for Contour Integration
Re: a), one prefers $e^{iz}$ to $\sin z$ or $\cos z$ because $e^{iz}$ goes to zero as $|z|\rightarrow\infty$ in the upper half-plane. This often gives one a way to use a semicircular contour, with radius tending to infinity, and argue that the circular arc does not contribute to the integral in that limit.
Apr
15
answered To prove this log inequality (middle school)
Apr
3
comment When flipping a coin, why is the expected time till 3 Tails in a row different from Tail-Head-Tail?
The difference is that an initial T-H makes you start all over when watching for T-T-T, while an initial T-T lets you make use of the pattern-breaking T as the start of the next chance at T-H-T.
Apr
3
comment Is there a difference between these two definitions of differentiable at a point?
And non-differentiable functions like $f(x)=|x|$ (at $x=0$) will satisfy the first definition as well.
Apr
3
comment Is there a difference between these two definitions of differentiable at a point?
Any smooth function will satisfy the first definition with any value of $f'(x_0)$ whatsoever; it doesn't pick out a unique value.
Apr
1
comment Is there a difference between these two definitions of differentiable at a point?
The point is that the difference between $f(x_0+\Delta x) - f(x_0)$ and $f'(x_0)\Delta x$ must go to zero rapidly. Your definition just says it goes to zero... but of course it will, as long as $f$ is continuous at $x_0$, and regardless of what you assign $f'(x_0)$ to be, because it's the difference between two vanishing quantities.
Mar
29
comment Find the interval of convergence for $\sum_{n=1}^∞ sin(\frac{1}{n}) tan(\frac{1}{n})x^n$
For large $n$, $\sin(1/n)\sim \tan(1/n) \sim 1/n$. Use the ratio test with this in mind.
Mar
25
comment Solve $x^n+y^n = (x+y)^n$
$n=1$. $x=0$. $y=0$. And $x=-y$ for odd $n$. That's it, right?
Mar
20
comment Guess the number. Maximizing expected winnings?
Presumably the man wants to minimize your expected winnings, as it's zero-sum. What is his optimal strategy?
Mar
18
revised Is there a closed-form expression for $\sum_{k=1}^{n}\lfloor k^{q} \rfloor$ for $q \in \mathbb{Q}_{> 0}$?
Fixed error in summation.
Mar
18
answered Is there a closed-form expression for $\sum_{k=1}^{n}\lfloor k^{q} \rfloor$ for $q \in \mathbb{Q}_{> 0}$?
Mar
18
comment Can a complex number be prime?
Pretty sure the Gaussian integers are a unique factorization domain. Maybe you're thinking of other field extensions of $\mathbb{Z}$, like $\mathbb{Z}[\sqrt{-5}]$?
Mar
15
revised Prime numbers are related by $q=2p+1$
deleted 27 characters in body
Mar
15
comment Prime numbers are related by $q=2p+1$
What does "it has the sum of numbers not exceeding $3$" mean?
Mar
14
comment Proof of $\pi$ not being a quadratic irrational number.
This doesn't answer the question... OP wants to know if there's a simple proof that $\pi$ isn't a quadratic irrational, i.e., an irrational root of a quadratic with rational coefficients.