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22h
answered Prove $\sum_{n=1}^{\infty}\frac{|\sin{(n\theta_0)}|}{n}$ diverges for given $\theta_0\in (0,\pi)$?
2d
comment Does $\sum_{n=1}^\infty \frac{\cos(\ln(n))}{n}$ converge?
The series is $(R,1)$ summable and this is indeed the limit in $(R,1)$ sense.
Sep
16
answered How can I simplify $ \sum_{r=0}^{m-1}r^3\frac{\binom{m}{r}(m-r)!\begin{Bmatrix} n\\ m-r \end{Bmatrix}}{m^n}$?
Sep
16
comment Prove the congruence $pB_{p-1} \equiv -1 \pmod p$ for Bernoulli numbers.
Actually, for your case, just put $j=p-1$. Then reduce mod $p$. The LHS is by Fermat's little theorem, $p-1$, and the RHS is $pB_{p-1}$.
Sep
16
comment Prove the congruence $pB_{p-1} \equiv -1 \pmod p$ for Bernoulli numbers.
This is a special case of Von Staudt Clausen theorem. Wikipedia has a proof.
Sep
16
comment How can we apply the Borel-Cantelli lemma here?
(+1) Nice! on mouseover tech.
Sep
16
comment As$\ n \to \infty$, how does a product over the primes less than$\ p_n$ equal the same product over the primes less than$\ n$?
I don't see the point that you had to ask the same question again, and it had already been answered there.
Sep
16
comment Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing?
I guess you should ask "convergence or divergence" of the series $\sum \frac{d(n)}{n\log^2 n}$
Sep
16
comment A Question From Measure Theory
It can be constructed from a measure-zero set.
Sep
16
comment A Question From Measure Theory
imsc.res.in/~sunder/mgnvss.pdf
Sep
12
comment A short question on the estimation of $\sum_{1\leq n\leq x} \mu(n)n^{-1}$.
By Prime Number Theorem, the sum coverges to $0$.
Aug
28
comment Limit of $S(x) = x − x^2 + x^4 − x^8 + x^{16} − x^{32} + \cdots$ as $x$ approached $1$ from below
I think 0.8 is not enough. 0.995 in the solution by Noam Elkies is already very close to 0.5.
Aug
28
comment Limit of $S(x) = x − x^2 + x^4 − x^8 + x^{16} − x^{32} + \cdots$ as $x$ approached $1$ from below
$S(x^4)\neq 0.8^4$
Aug
28
comment Limit of $S(x) = x − x^2 + x^4 − x^8 + x^{16} − x^{32} + \cdots$ as $x$ approached $1$ from below
There's a way without numerical observation.
Aug
24
comment Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
@Did I do not know about this set, but the set of initial points such that the sequence bumps into $\pi/2+\pi\mathbb{Z}$ should be countable.
Aug
24
comment Calculate $\int_{S^2}\frac{1}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}}dS$ where $a^2+b^2+c^2<1$.
Shifting the origin without affecting the integral should be justified. Otherwise, your calculation is just when $a=b=c=0$.
Aug
24
comment Calculate $\int_{S^2}\frac{1}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}}dS$ where $a^2+b^2+c^2<1$.
$x,y,z$ are not parametrized that way.
Aug
24
comment Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
Maybe it would be possible to get the set of initial points that makes the sequence dense in $\mathbb{R}$. But, even that seems extremely difficult.
Aug
23
answered Using Borel-Cantelli Lemma
Aug
23
comment Evaluation of the series $S(\omega)=\sum\limits_{k=0}^\infty (-1)^k {\alpha \choose k}\cos(k\omega)$
@ecook As far as I know for $0<\alpha<2$ as in your assumption, the power series converges for all $|z|\leq 1$, $z\neq 1$. For $z=1$, the analysis should be more careful, but I did not work it out yet. Also, there would be more to be said about $\alpha$. Such as, what are the set of all $\alpha$'s such that $\sum_k\binom{\alpha}{k} z^k$ converges for all $|z|\leq 1$, $z\neq 1$?