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z8KITt+@^GKVM<0N&aqT~xeA34dTdm*nqLBEwCyhQ0oii&!JX#_GH2g!-rHE*D5_6F7M zTjVnw57viV;}j0vKYyz3Fgtr`oHf+}nGa*qy(ktDPid!QJ#KUQT&W8(+Rfr>eo7WF zYcD1lQozA)gGZMM6@yK`QRjA%XR@*p1rfz{GrfaC{lTWUOIL3RyT+3G3euT-y*@ae zESKvn?0F2LteSiRpJb=P7G8g&C)RLbg6Sj?5^HMw-|OE!_@@jx4Ew@W;?UgdEphQr zl~v2Y09A){01u1ZhGd|e>%TJSVltHyu^c(&cBt(MP9?9@A>Ol$zY@h0%>YdH?s7|k ze~05M#EJYVF=f-fYVc<%uOruoKfTHZf>u}1XfkHiYK76x%6&=vG%%h*% zH2S*rct=?hFqkwTFV@*b`j1N*@J;jF)mk5t`A1`F(K~n+nl|i%8sAbpfrKo;C`$)U z8~P4DTHu1l-hps~Jtn>0QE|(KDxM@D-D<=VSf%X`uhaGZZX}HLFba3NtnI$f3Ca7P zR&K=r*fDO<@7eAD0`UJm92+3{71P`t`10Ss>VIwYj~Y98t$`h-L4QB~PXiXJg20AE k1O@th*Zp6w{y$yHQ;ZzTSvSEC>y&`Y!_}8k;X-5o7g&!OAOHXW literal 0 HcmV?d00001 diff --git a/15e640-twitter_post_sqrt_root_limit/main.typ b/15e640-twitter_post_sqrt_root_limit/main.typ new file mode 100644 index 0000000..27a3108 --- /dev/null +++ b/15e640-twitter_post_sqrt_root_limit/main.typ @@ -0,0 +1,33 @@ +Notes on this post: + +$ + 1=lim_(n->0)sqrt(n+sqrt(n+sqrt(n+ ...))) +$ + +Some argues it's $0$ or $1$, that's simply because the form is ill-defined and the post is misleading intentionally, here's another way to look at this: + +define $f_m$ as a series of functions: + +$ + f_m (x) = underbrace(sqrt(x+sqrt(x+sqrt(x+...))), "m times") +$ + +In this form we have two ways to write the original limit: + +$ + lim_(n->0) lim_(m->oo) f_m (n) +$ + +or + +$ + lim_(m->oo) lim_(n->0) f_m (n) +$ + +Limits are not interchangeable by default, and orders matter, when $display(lim_(n->0) f_m (n))$ comes first it's undoubtedly $0$, when $display(lim_(m->oo) f_m (n))$ comes first it's $1$. + +Here's how $f_m (x)$ looks like with $m=1,2,3,4,5$: (Credit: WolframAlpha) + +#image("image.png") + +I hope this clears things up. \ No newline at end of file diff --git a/2bc0c8-2024_spring_TA/green.typ b/2bc0c8-2024_spring_TA/green.typ new file mode 100644 index 0000000..5dfc086 --- /dev/null +++ b/2bc0c8-2024_spring_TA/green.typ @@ -0,0 +1,11 @@ +$ +integral.double_(D) v laplace u - u laplace v dif x dif y = integral.cont_(diff D) v (diff u)/(diff bold(n)) - u (diff v)/(diff bold(n)) dif s +$ + +$ +integral.double laplace u dif x dif y = integral.cont (diff u)/(diff bold(n)) dif s +$ + +$ +nabla = bold(e)_r (diff)/(diff r) + bold(e)_theta 1/r (diff)/(diff theta) + 1/(r sin theta) bold(e)_phi (diff)/(diff phi) +$ \ No newline at end of file