@SimplyBeautifulArt I've seen some explanations of it before, but I don't see the point of it. I know some physicists claim that it has physical implications, but as of now I weigh the evidence at about 90% confidence that they are talking nonsense.
I know, but what do you mean by regularize? There is no one single definition.
You can arbitrarily choose some kind of cutoff function as a so-called regularizer, and then it just comes down to a question of whether some other summation converges.
@SimplyBeautifulArt But as you know there are many many different methods of assigning values to divergent series that extends standard convergent summation, and they don't even always agree.
So what is your goal? Clearly you want a new assignment that has some properties other than what you've mentioned, namely agreement on convergent series.
@SimplyBeautifulArt you might be interested in reading about functional analysis for applications to numerical approximation techniques. They go hand in hand together.
@SimplyBeautifulArt −3.219195 Unfortunately, attempting your challenge made me find out that Graph cannot perform numerical integration to high precision, so I in fact have no choice but to use series acceleration to do it...
Actually I give up. I cannot be certain of the last digit "5" using my software.
I could plug into WA though.
Hmm WA doesn't allow more than 5dp???
It seems that series converges too quickly for series acceleration to be accurate!
Well the trivial one; let g(x) be the definite integral from 0 to x, and h1(x) = (g(x)+g(x+1))/2, and h2(x) = (h1(x)+h1(x+1))/2 = (g(x)+2·g(x+1)+g(x+2))/4, and so on. h2 is too slow. h3 and h4 converge fast but not fast enough before Graph get numerically inaccurate.
Using the contour
contour integration gives
$$
\begin{align}
\overbrace{\color{#C00}{\text{PV}\int_0^\infty\frac{e^{i\pi x}}{\log(x)}\,\mathrm{d}x}}^{\substack{\text{integral along the line}\\\text{minus an infinitesimal}\\\text{interval centered at $1$}}}+\overbrace{\vphantom{\int_0^\infty}\ ...
Thanks! Aha I was right in doubting my last digit!
I'm sure I can find a purely numerical method for such quickly converging integrals. I don't quite like special methods that only work for special cases hahaha..
@SimplyBeautifulArt: I'm beginning to suspect that perhaps it's the Graph software that has inaccurate numerical integration, rather than the acceleration method. Do you mind computing h4(100) and h4(101) in any other software you have?
Which means you just need the 6 definite integrals over [0,n] for n in [100..105].
@user21820 :-/ Well if the series is alternating and absolute value of terms are monotonically decreasing, then its obviously bounded towards the limit.
What I mean is that the method won't allow you to numerically compute limits of jerkily converging alternating series to high precision, despite the theoretical limit being preserved by the acceleration.
Assuming linear rate of convergence usually works well for monotonically converging sequences (unless it actually converges faster than that).
Will you share with me now what your class schedule(s) are this term? Last I asked, they were yet to be confirmed. Of course, no need to share all, but what math class(es) are you enrolled in this fall? (Again, only if you want to share.)
@amWhy Oh, sure. I'm taking AP Physics Electro-Magnetism, AP Macroeconomics, AP Computer science A, AP US gov, chinese, and then some English + electives involving the use of some Adobe products
Well, I still believe that learning to write in any language is best started with a pencil (or pen) and paper. I know Spanish, but can utilize on my computer the entire alphabet, and in particular, accents! But it is doable. So I am quite intrigued about Chinese.
Using the contour
contour integration gives
$$
\begin{align}
\overbrace{\color{#C00}{\text{PV}\int_0^\infty\frac{e^{i\pi x}}{\log(x)}\,\mathrm{d}x}}^{\substack{\text{integral along the line}\\\text{minus an infinitesimal}\\\text{interval centered at $1$}}}+\overbrace{\vphantom{\int_0^\infty}\ ...
This is the Realm of Simply Beautiful…
This is the Realm of Simply Beautiful…