@robjohn nothing as such ... the construction is mind blowing !! I don't think I would have learnt such a construction if you proved the result directly (instead of the contradiction proof) :-)
@robjohn functions that satisfy that property seems interesting ... does it have deeper properties ?
Some intuition why your proof isn't quite correct: When you used the triangle inequality
$$|(-1)^{n+2}a_{n+1} + \cdots + (-1)^{n+k'+1}a_{n+k'}| \leq |(-1)^{n+2}a_{n+1}| + \cdots + |(-1)^{n+k'+1}a_{n+k'}|\\
= a_{n+1} + \cdots + a_{n+k'} $$
you removed the alternating nature of the series, so if y...
@Sawarnik In case you still care :P I added an alternative proof (based on trig ineq) here .. I also proved a few related Jack Garfunkel's inequalities .. tell me if you are interested in the proofs :D :)
@Chris'ssis @robjohn I have a nice limit problem :D $$\lim\limits_{n\to\infty} \frac {n}{\ln^2 n}.\left[\left(\sqrt[n]{H_n}\right)^{H_n}-1\right]$$, where $H_n = 1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}, n \ge 1$ :)
@r9m, well, it was easy to see that the log went to infinity, and the harmonic series goes to 1, and the '-1' made it zero. so 0/infinity = 0. l'hopital's rule i think?
i split it up like this : $$\lim_{n\to \infty } \, \frac{n \left(\lim_{n\to \infty } \, \left(\left(H_n\right){}^{1/n}\right){}^{H_n}-1\right)}{\log ^2(n)}$$
i love the harmonic series.. here's a fun identity i noticed the other day : $$e^z=-1+\frac{2}{1-\frac{\pi z}{-i z H_{\frac{i z}{2 \pi }}+i z H_{-\frac{i z}{2 \pi }}+2 \pi }}$$
@DavidKirby I am not writing a paper, no. But I'd have to do that soon.
Well, here's what I did, in short :
A quintic is not solvable by radicals, which comes from the basics of Galois theory. What I proved is that the Poincare autforms satisfy a general quintic, which comes from the fact that a certain algebraic curve $X(5)$ "looks like" a Riemann sphere with exactly 12 points removed from the vertices of some circumscribed icosahedra.
So, strictly speaking, the riemann surface of $x^5 - x - Z$ "looks like" the curve $X(5)$, which is quite interesting.
Alongside, I derived a complete different proof of the exceptional iso $A_5 \cong PSL_2(\Bbb F_5)$ and solved the inverse galois problem for $PSL$ groups.
@DavidKirby Thanks. I suspect that no abstract proof of solvability of quintics over function fields $\Bbb C(j)$ has been stated anywhere (though this fact was known for years), but even if it is not the case, my understanding increases 10fold and it was an extreme pleasure to discover this wonderful connection between algebra and geometry.
@Balarka Sen, specifically i was wondering about the inverse of $x^5-x-Z$, but then i realized that 'automorphic' was the word i was looking for
@Balarka Sen, but yeah, no need to explain further. i need to find the right questions before getting the right answers, so reading up on the whole shebang at the moment. ;)
@DavidKirby Yeah, well, that's the whole point, proving that the inverse is algebraically dependent of poincare automorphic forms. What is more, one can transform the quintic into a sextic. This is a consequence of $A_5 \cong PSL_2(\Bbb F_5)$ (the former acts of 5 points and the later on six $\{0, 1, 2, 3, 4\} \cup \infty$.
This sextic is the desired surface we look for. Indeed, it's the modular equation for $X(5)$ and is parameterized by ellmod functions.
@DavidKirby haha, i don't think it has enough importance to be published but will see.
first i have to show my calculations to a few people like my professor and professor V. Kumar Murty, who proposed the problem and asked me to write it up.
@Balarka Sen, I often wonder about the results that seem unimportant or not worth publishing -- a web app where those could be published and reviewed/refined/appropriated/etc could be pretty neat.
One person's statistically insignificant regression coefficient is another person's treasured additive white Gaussian noise.
How do I calculate historical yearly market worth knowing only its yearly growth rate and value for year 2015?
"According to MarketLine, the global home improvement market is expected to grow at 2.5% yearly rate between 2010-2015 and is predicted to be worth almost $678 billion by 2015"
That would be regular algebra, generally, you can define 2015 as your 0 point for X axis, and then you get the function $f(x)=1.025x+678$ considering every point on y axis is in billions
If you just want the year values, recall it grows 2.5% yearly, so you can have the 'magic square' to do the rest of the work for you. But it's the same as the function.
How do I calculate historical yearly market worth knowing only its yearly growth rate and its martket worth for year 2015?
Example:
According to MarketLine, the global home improvement market is
expected to grow at 2.5% yearly rate between 2010-2015 and is
predicted to be worth almost $6...
@Studentmath I am making presentation not for someone specific but I want it to be on SlideShare website. I only have yearly 2.5% and 2015 economical data. If I calculate to discover historical economical data, will it be legit?
@r9m I think I also saw it here some time ago arxiv.org/pdf/1105.0859.pdf. I didn't search for it now, but I think it is there or something very similar.
@r9m There you have a bunch of awesome inequalities. By the way, that integral inequality I showed you some time ago only uses the elementary stuff in high school. No need for powerful inequalities.
Your "art", integral and series computation, is fun but not that much important in mathematics, right? =)
By "important" I don't mean useless, but rather an "important tool". Integrals and series closed forms are used, but not frequently as they aren't usually a good tool in general branches of mathematics.
I'd actually look for global function before looking at local ones. $$\int f(x)g(x) \mathrm{d}x = \left ( \int f(x) \mathrm{d}x \right ) \left ( \int g(x) \mathrm{d} x \right )$$
@Chris'ssis we could go for finding $O(R(n))$, where $ \displaystyle R(n) = 1 + \frac{e^n.n!}{2n^n} - \frac{n!}{n^n}\left( 1+ \frac{n}{1!}+ \cdots +\frac{n!}{n!} \right)$ :-) .. but thats like tracing back robjohn :P
How do I calculate historical yearly market worth knowing only its yearly growth rate and its martket worth for year 2015?
Example:
According to MarketLine, the global home improvement market is
expected to grow at 2.5% yearly rate between 2010-2015 and is
predicted to be worth almost $6...
