/sci/ - Science & Math

Once upon a time, Euler investigated a particularly interesting infinite series:
<span class="math">\ldots + x^3 + x^2 + x^1 + 1 + \frac{1}{x^1} + \frac{1}{x^2} + \frac{1}{x^3} + \ldots[/spoiler]
This was before a theory of infinite series was totally developed. We could ask ourselves what this series might sum to. For instance, if we multiplied the series by <span class="math">x[/spoiler] then we'd just get the series back again. If we call the sum <span class="math">S[/spoiler] then
<span class="math">Sx=S[/spoiler] and then <span class="math">Sx-S=0[/spoiler] so mysteriously
<span class="math">S(x-1)=0[/spoiler]
Which would mean, unbelievably, that this sum is zero whenever <span class="math">x \neq 1[/spoiler].

Let's look closer. For instance, suppose we multiply this series by 9 when <span class="math">x=10[/spoiler]. We then have the strange number <span class="math">\ldots 999.999 \ldots[/spoiler]. Nowadays I suppose we all have learned or heard that <span class="math">0.999\ldots = 1[/spoiler] but then our reasoning would suggest that <span class="math">\ldots 999 = -1[/spoiler].

(continued)

Between the right systematically destroying science and education and the left bending over to give poor people money instead of NASA, the United States is officially going to be a shithole in 20 years.

Where do I move to for the sake of science? I love science and think it is critical for our species to do as much of it as possible. Which country is pro-science?

Pic related. It's science.

A pretty young thing at work here was talking about her boyfriend, and she said, "He's a nice guy. I only date nice guys." I said, "For every girl who is dating a 'nice guy' there's a nice guy she's *not* dating who thinks she only dates assholes."

I'm pretty sure this is scientifically true, but I was wondering if anyone has any data to add to the study.

Ok, I understand the Laplace equation for Cartesian coordinates.
<div class="math"> \nabla^{2}T=0 </div>

<div class="math"> \frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial y^{2}}+\frac{\partial^{2} T}{\partial z^{2}}=0 </div>
My question is how do you derive the solution for cylindrical and spherical coordinates?
<div class="math"> \frac{1}{r}\frac{\partial^{2} T}{\partial r^{2}}+\frac{1}{r^{2}}\frac{\partial^{2} T}{\partial \theta ^{2}}+\frac{\partial^{2} T}{\partial z^{2}}=0 </div>

<div class="math"> \frac{1}{r^{2}}\frac{\partial }{\partial r}(r^{2}\frac{\partial T}{\partial r})+\frac{1}{r^{2}sin(\theta )}\frac{\partial }{\partial r}(r^{2}\frac{\partial T}{\partial r})+\frac{1}{r^{2}sin^{2}(\theta )}\frac{\partial^{2} T}{\partial \phi^{2} }=0 </div>
Everywhere I look mostly just states the upper two equations as fact, how do you actually arrive at them?

>>2431527

Well kids, I grabbed "The Singularity is Near" yesterday on a whim. Am I in for wild, Kaku-style fantasies of futurists, or something realistic?

Pic unrelated, but science.

because we cant prove multiple dimensions/ multiverses

superior