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8h
comment Jordan measure problem
Why don't you prove it? Or at least show what you tried.
1d
revised a certain simple continued fraction
added 111 characters in body
1d
comment a certain simple continued fraction
Nicco -- It seems now that if you just use the sums of your geometric series for the convergents, (where you have already let $k \to \infty$) then you get $C(n)=F_n/(F_{n+1}-F_nq)$ for the $n$th convergent. If the top and bottom here are divided by $F_n$ it is seen to approach $1/(\phi-q)$ as I noted in a comment to my answer. This is only off by missing a $\phi$ from your formula, since it can also be written as $\phi / (1+\phi-\phi q).$
1d
comment a certain simple continued fraction
Let us continue this discussion in chat.
2d
comment a certain simple continued fraction
@Nicco -- I don't follow the last comment. The cfrac has no position in which to replace $\phi$ by various things, only your final formula has $\phi$ in it. In the cfrrac itself there are only the terms $1-q,1-q^{3^k},$ and so on, As $k \to \infty$ all of these except for the first one, with no $k$ in it, will approach $1$ making the whole thing go to $[0,(1-q),1,1,...]$ or $1/(1-q+\phi-1)=1/(\phi-q),$ which for $q=1/2$ is $0.89442...$.
2d
comment a certain simple continued fraction
@Nicco See the last added part of my answer, I think for k to infinity the cfrac is still somewhere between 2/3 and 2.
2d
revised a certain simple continued fraction
added 1285 characters in body
2d
comment a certain simple continued fraction
@Nicco If you mean to take the limit of your continued fraction as $k \to \infty,$ then that should be stated in the posted question. As it seems, the question claims a result independent of $k.$
2d
revised a certain simple continued fraction
corrected convergent numerator
2d
answered a certain simple continued fraction
2d
comment a certain simple continued fraction
I think one does need to say "iff $k \in \mathbb{N}$" somehow, provided it is intended that the equality be false when $k$ is not a natural number.
2d
answered Proving $\sin^2(x) + \cos^2(x) =1$ using calculus
Aug
29
answered Minimum of $f(x)=\sum_{i=1}^n\frac{a_n}{x-b_n}$ occurs at extreme point?
Aug
26
comment A particular polynomial - 2
Let us continue this discussion in chat.
Aug
26
comment A particular polynomial - 2
@Turbo Did you notice that in the first paragraph is a proof it cannot be done with all positive terms? Also what is your view on terms with irrational coefficients? And by Z[I] coefficients do you mean coefficients in the Gaussian integers? If so since there is no order relation there it may be possible, since 2 has a squareroot in a sense namely $1+i$.
Aug
26
comment A particular polynomial - 2
@Turbo I noticed your apparent restriction to integer coefficients. Maybe this could be done, but my example has an irrational coefficient in the factors.
Aug
26
comment A particular polynomial - 2
@Turbo Just added an example fitting your list, only four nonzero products of two squares.
Aug
26
revised A particular polynomial - 2
added 349 characters in body
Aug
26
answered A particular polynomial - 2
Aug
26
comment A particular polynomial - 2
Turbo: OK but one part is a negative result, and my other part gives one having all terms including the other two products of squares. Post soon...