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Another layman blackhole question, pulling one end of a string out from behind the event horizon No long explanation is needed,  What would happen if I were to allow one end of a rope to fall past the event horizon of a black hole while I held the other end?  Would I be able to pull it out? Would the rope feel extremely (infinitely?) heavy? Radius of Convergence of power series of Complex Analysis I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be greatly appreciated!  (i) $\displaystyle\sum_{n=0}^\infty \bigg(\dfrac{1}{2n}\bigg)z^{3n}$.  (ii) $\displaystyle\sum_{n=0}^\infty(2^n+i)(z-i)^n$.  (iii) Show that the radius of convergence of $\displaystyle\sum_{n=0}^\infty z^{n!}$ is $1$ and that there are infinitely many $z \in \mathbb{C}$ with $|z|=1$ for which the series diverges. $A$ and $B$ are $n\times n$ matrices and $A+B=AB$. I have an interesting proof that this implies $A$ commutes with $B$, but the proof only works when $||B|| \lt 1$. I'm looking for a way to salvage the proof so that it works for all $B$.  $$A=AB-B$$ $$A=(A-I)B$$ by repeatedly substituting $A$ with $(A-I)B$ on the RHS, $$A= (A-I)B^N - \sum_{i=1} ^{N-1} B^i$$  Since the space of $n\times n$ matrices is Banach (hence complete) under the standard operator norm, the limit $\sum_{i=1}^{\infty}B^i$ is well defined given $||B|| \lt 1$. Hence, taking the limit as $N \rightarrow \infty$, $$A =  -\sum_{i=1}^{\infty}B^i$$  Now the LHS commutes with B, hence so does the RHS.    Is there some trick to extend the method to work with any $B$ where $||B|| \geq 1$? I know that the result is true due to another simple algebraic proof, but I'd like to see if this method can be salvaged. lists Ubuntu 4.10 (Warty Warthog) as the first Ubuntu released by Canonical Ltd.  But I wanted to know whether there ever was an Ubuntu 1.x, as in Windows or Mac OS. If yes, is it available somewhere? Plotting function defined with Lua doesn't do anything From 8th June there's been a  period for people entering the UK. From 10th July, travellers from  via a travel-corridor. If more countries are added to the list of exemptions then the date for each country will be later than the 10th July. If a traveller entered the UK from a country that is not on the exempt list and that country is added to the exempt list prior to the end of the 14 day self-isolation period does the self-isolation period end early? For example, if an individual entered the UK from France on the 5th July, should they remain in isolation for 14 days or does the isolation end on the 10th when France is added to the list of exempt countries? How do I create a reflective white material? I'm pretty new to this and this is my first question here, any help would be greatly appreciated.  So basically what I want to achieve is a floor material as seen in this photo...    ...the reflections are very crisp but dim.  With a glossy shader I get the following results:    First picture: reflections are blurry  Second picture: roughness of the material looks right but reflection is bright red  Third picture: If tinted in any other color than white, it seems to produce the result I want. Reflection is sharp and gets tinted blue. What I need is the reflection to be tinted white.  Thanks for any help. On Wikipedia, it says that $\lim_{n\to\infty}(1+\frac{1}{n})^n = e$ :     It [e] is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, ... ()   When I evaluate $(1+\frac{1}{n})^n$ for $n = 10^8$, I get approximately $2.718281798347$ which indeed is pretty close to $e$.  But when I try to "solve" the limit using the laws of limits, I get  $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = \left(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)\right)^n$$  because of the power law  $$ = \left(\lim_{n\to\infty}(1)+\lim_{n\to\infty}\left(\frac{1}{n}\right)\right)^n$$  because of the addition law  $$=\left(1+0\right)^n = 1$$  but that would mean that $e=1$, which is obviously not true.  What am I missing / doing wrong?  Thanks in advance. I just updated my iMac to Mojave, and am still trying to setup my preferences.  I tried dark mode, but did not like it (half of me apps seem to ignore it anyway).  I switched to light mode, but this too has problems. The File Save dialog shows a list of Directories and existing files, but the latter are very light grey, almost indistinguishable from the background.    I have searched the Preferences menu, but can't seem to find any setting to set normal contrast levels like in High Sierra. I recently upgraded to Ubuntu 20.04 lts from 18.04 lts. After reboot audio is not working. I tried to reinstall pulseaudio and alsa but still not working. sudo apt-get install --reinstall alsa-base pulseaudio sudo alsa force-reload still not working. output  └─▶ inxi -SMA System:    Host: Wizard Kernel: 5.4.0-26-generic x86_64 bits: 64 Desktop: Gnome 3.36.1 Distro: Ubuntu 20.04 LTS (Focal Fossa)  Machine:   Type: Portable System: Dell product: Vostro 2520 v: N/A serial: &lt;superuser/root required&gt;             Mobo: Dell model: 0V1NDP v: A11 serial: &lt;superuser/root required&gt; BIOS: Dell v: A11 date: 02/24/2014  Audio:     Device-1: Intel 7 Series/C216 Family High Definition Audio driver: snd_hda_intel             Sound Server: ALSA v: k5.4.0-26-generic     output └─▶ lspci  00:00.0 Host bridge: Intel Corporation 3rd Gen Core processor DRAM Controller (rev 09) 00:02.0 VGA compatible controller: Intel Corporation 3rd Gen Core processor Graphics Controller (rev 09) 00:16.0 Communication controller: Intel Corporation 7 Series/C216 Chipset Family MEI Controller #1 (rev 04) 00:1a.0 USB controller: Intel Corporation 7 Series/C216 Chipset Family USB Enhanced Host Controller #2 (rev 04) 00:1b.0 Audio device: Intel Corporation 7 Series/C216 Chipset Family High Definition Audio Controller (rev 04) 00:1c.0 PCI bridge: Intel Corporation 7 Series/C216 Chipset Family PCI Express Root Port 1 (rev c4) 00:1c.3 PCI bridge: Intel Corporation 7 Series/C216 Chipset Family PCI Express Root Port 4 (rev c4) 00:1c.5 PCI bridge: Intel Corporation 7 Series/C210 Series Chipset Family PCI Express Root Port 6 (rev c4) 00:1d.0 USB controller: Intel Corporation 7 Series/C216 Chipset Family USB Enhanced Host Controller #1 (rev 04) 00:1f.0 ISA bridge: Intel Corporation HM75 Express Chipset LPC Controller (rev 04) 00:1f.2 SATA controller: Intel Corporation 7 Series Chipset Family 6-port SATA Controller [AHCI mode] (rev 04) 00:1f.3 SMBus: Intel Corporation 7 Series/C216 Chipset Family SMBus Controller (rev 04) 07:00.0 Network controller: Broadcom Inc. and subsidiaries BCM43142 802.11b/g/n (rev 01) 09:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 07) Strange book where teen protagonist is sent to a warzone, uses illusory magic to grow male genitalia SOAP or REST for Web Services? I have two questions for you guys. I really appreciate your help.  1.- Is there a way to rename an Availability Group?  2.- Is there a way to get the T-SQL statements in order to re create a Availability Group in case that I need to drop it and create it again?  If there's information that you want me to share, please let me know and I will provide it. How can I prove the maximum number of edges? On my book it says that the maximum number of edges E, in a simply connected undirected unweighted graph $G(V,E)$ is $\dfrac{|V|(|V|-1)}{2}$.  How can I show that it is true? Thanks! When can we use 'much' in affirmative clauses? Enabling GPU rendering for Cycles? I want to enable GPU rendering, but there is no option in User Preferences > System:    Why is this? How can I get cycles to render using my GPU? A non orientable closed surface cannot be embedded into $\mathbb{R}^3$ Can I programmatically open the devtools from a Google Chrome extension? doing comparison if else in JasperReports Probability of complete coupon set in $K$ boxes where a box has $N$ distinct coupons? Say there is a contest to collect a full set of $M$ coupons.  Each box of product has $N$ distinct coupons of the $M$ possible coupons in it, selected uniformly without replacement from the $M$ possible coupons, the same $N$ in number for all boxes.  I'm interested in the probability that after opening $K$ boxes of product I have at least one complete set of the $M$ coupons.  My attempt has been to take the probability of a coupon not being in a box, use that to determine the probability of that coupon not being in any of the $K$ boxes, and then... I get flummoxed, and don't think this is the correct approach.  I searched the site for coupon collector problems, there are a couple that deal with similar situations (multiple in a box), but with the box having coupons selected with replacement, so duplication in a box is possible.  Right now I'm using a CAS to get probability of sums of hypergeometric samples having all bins covered to check my ideas, but that's utterly impractical for other than tiny cases.   Is this not as trivial as I think it is, or is there a direct way to calculate this?