@robjohn that is true .. but I thought we were condensing the sum between $\left(2n-\frac12\right)\pi$ and $\left(2n+\frac32\right)\pi$ .. its $<0$ coz there are less $H_k$'s between $\left(2n-\frac12\right)\pi$ and $\left(2n+\frac12\right)\pi$ than there are between $\left(2n+\frac12\right)\pi$ and $\left(2n+\frac32\right)\pi$, so the later part sweeps more area under the curve $\cos x$ than in the previous interval ..
@meer2kat only my modstars are glowy and happy. the others are black and full of misery. you wouldn't want to do that to some poor string of random messages would you?
@meer2kat only my modstars are glowy and happy. the others are black and full of misery. you wouldn't want to do that to some poor string of random messages would you?
@robjohn we started with assuming $\sum\frac{\sin(H_k)}{k H_k}$ converges, then showed it means $\sum\frac{\cos(H_k)}{(k+1)}$ should converge .. if the later converges then $\sum\frac{\cos(H_k)}{(k+1) H_k}$ converges .. how do we show this one diverges ?
If anyone has comments on my reading list (books I should remove, books I should add, books I should change the order of) I'd highly appreciate it: pastebin.com/xZews1wM
(The Algebraic Geometry section is currently empty because I haven't quite decided what to do there)
Maybe the Homological Algebra stuff should also go before it, actually.
@r9m what do you mean. I showed that if $\sum\frac{\sin(H_k)}{kH_k}$ converges, then $\sum\frac{\cos(H_k)}{k+1}$ converges, but it doesn't. Then we considered whether $\sum\frac{\cos(H_k)}{k+1}$ was even bounded, and I then said if it were, then $\sum\frac{\cos(H_k)}{(k+1)H_k}$ would converge, but that is the same as $\sum\frac{\sin(H_k)}{kH_k}$ converging, so we can't have that $\sum\frac{\cos(H_k)}{k+1}$ is bounded.
@PedroTamaroff sorry... I didn't mean to delete your comment :-(
@FernandoMartin lol, no. Spectral sequences were developed in the 40's, and are pivotal wherever homological is (e.g. algebraic geometry and complex geometry). Everything now is super combinatorial/categorical, e.g. operads.
Some of them were smiling at me, but we'll see, @Pedro ... some proofs and integral/form computations, plus linear algebra with change of basis and eigenvectors, etc.
I gave my kids on the final today a question that was: Compute $\int_C \omega$, where $\omega=df$ and $C$ is a path joining any two points on the surface $f=5$.
@seaturtles You take $F=Z(D)$ and look at $D$ has a vector space over $F$; then do some calculations with cyclotomic polynomials, the class equation in $D^\times$ and so on.
