Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for $n=2,3,...$. Define $$P_n=\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)...\left(1+\frac{1}{a_n}\right).$$ Find $\lim\limits_{n\to\infty} P_n$.   My approach: $$P_n=\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)...\left(1+\frac{1}{a_n}\right)$$ $$=\left\{\frac{(a_1+1)(a_2+1)...(a_n+1)}{a_1.a_2...a_n}\right\}$$ $$=\left\{\frac{1}{a_1}\left(\frac{a_1+1}{a_2}\right)\left(\frac{a_2+1}{a_3}\right)...\left(\frac{a_{n-1}+1}{a_n}\right)(1+a_n)\right\}$$ $$=\left\{\frac{1}{a_1}.\frac{1}{2}.\frac{1}{3}.....\frac{1}{n}(1+a_n)\right\}$$ $$=\frac{1+a_n}{n!}, \forall n\in\mathbb{N}.$$   Thus, we need to find $$\lim_{n\to\infty} \frac{1+a_n}{n!}.$$ How to proceed after this?
Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for $n=2,3,\dots$. Define $$P_n=\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)\dots\left(1+\frac{1}{a_n}\right)$$Find $$\lim_{n \to \infty}P_n$$   Trial: $$1+\frac{1}{a_n}=1+\frac{1}{n(a_{n-1}+1)}\;.$$ But I can't simplyfy.Please help.
I installed Ubuntu 15.04 on the VM and I have downloaded Eclipse Luna. When I try to install it, this error occurs:      An error has occurred.   See the log file   /home/master/.eclipse/1473617060_linux_gtk_x86_64/configuration/1442224462970.log   Can you help me to fix  it? Thanks Is there an easy way to disable this laptop's touchpad? Specifically, I'm on a . I'd like to disable the  so that I can re-enable if my mouse were to run out of batteries (so physically breaking the connection to the motherboard or uninstalling the driver will not be the best options). In the Control Panel, when I go to "Mouse" and find the touch pad, there is no disable option. Any ideas?  Edit: I am on Windows 7. There is a driver on the Dell site that might allow an option to disable it, but I cannot install it on Windows 7. Differences between the introduction of a sandwich thesis and a paper I would like to know what are the main differences between the introductory part of a paper and of a thesis. My thesis would be a sandwich type, so I would have to wrap everything there.  Some of the differences are:   Longer in thesis. Deeper in thesis. Extended definition of key terms, concepts and methodology in thesis. Clear definition of the whole goal in thesis, whereas paper smaller goal.   To narrow the possible answers, I do research on the molecular aspects of the infection steps of a pathogenic swine bacteria. Also develop strategies to block infection by using vaccines. Can I use a HDD on my PS3 if I have already used another on the same console? I'm thinking of changing the Hard Disk of my PS3 (Slim 120GB), but I am unsure of what will happen to the data on the one I previously had if I do this:   PS3 has some data in the HDD. I turn off PS3, change HDD, install OS on new HDD. I download some data on new HDD from PS Plus.   Let's imagine I forgot to back something up...   I turn PS3 off, change HDD back.   Can I still use the old hard drive without needing to format it again? Why didn't the machine affect Bond in Spectre? How does one prove the following limit? $$   \lim_{n \to \infty}   \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}} = 3. $$ I have been racking my brain to understand the solutions of constrained stars and bars problems like:  $a+b+c+d = 19$;  $a,b,c,d \in \{0,1,2,3,4,5,6,7\}$  Finding the total number of solutions is really simple $\dbinom{22}{3}$.   What to do after that? I want to understand the entire procedure step by step.   Please do not mark it as a duplicate of .  Reasons:   The accepted answer's trick doesn't always work.   Another person (the answer with two upvotes) has just listed out the binomial coefficients and I can't even remotely understand what he has tried to do.   The second answer isn't my cup of tea. Why do people buy stocks that pay no dividend? Before recently, Apple Inc. did not give dividend to shareholders.  This makes me curious why people buy Apple Inc. stocks; while it might be due to the expectation of selling stocks at a higher price, this can only be possible if other people are thinking the same, thereby increasing demand. But the stocks themselves do not pay interests - so Apple's amazing revenue/profit records would not matter.... ffmpeg is not even cutting this and other mp4s to the nearest second I have a file  Other - Analogue Clock-JEJqy1Wlovw.mp4  which is just a download from     It's 2 minutes 13 seconds long.  I had some issues cutting it to the second.  I was able to use this command   ffmpeg -ss 0 -i infile.mp4 -c copy -t 60 output.mp4  to make a file that was 60 seconds, so no problem so far.  That's the file exactly 1 minute, no issue there so far.  C:\vids\a&gt;dir blahhclo.mp4   9,590,540 bytes  C:\vids\a&gt;   C:\vids\a&gt;mediainfo blahhclo.mp4 General Complete name                            : blahhclo.mp4 Format                                   : MPEG-4 Format profile                           : Base Media Codec ID                                 : isom File size                                : 9.15 MiB Duration                                 : 1mn 0s Overall bit rate                         : 1 278 Kbps Writing application                      : Lavf58.3.100  Video ID                                       : 1 Format                                   : AVC Format/Info                              : Advanced Video Codec Format profile                           : High@L4.0 Format settings, CABAC                   : Yes Format settings, ReFrames                : 3 frames Codec ID                                 : avc1 Codec ID/Info                            : Advanced Video Coding Duration                                 : 1mn 0s Bit rate                                 : 1 146 Kbps Width                                    : 1 920 pixels Height                                   : 1 080 pixels Display aspect ratio                     : 16:9 Frame rate mode                          : Constant Frame rate                               : 25.000 fps Color space                              : YUV Chroma subsampling                       : 4:2:0 Bit depth                                : 8 bits Scan type                                : Progressive Bits/(Pixel*Frame)                       : 0.022 Stream size                              : 8.21 MiB (90%)  Audio ID                                       : 2 Format                                   : AAC Format/Info                              : Advanced Audio Codec Format profile                           : LC Codec ID                                 : 40 Duration                                 : 1mn 0s Bit rate mode                            : Constant Bit rate                                 : 126 Kbps Channel count                            : 2 channels Channel positions                        : Front: L R Sampling rate                            : 44.1 KHz Compression mode                         : Lossy Delay relative to video                  : 40ms Stream size                              : 920 KiB (10%)    C:\vids\a&gt;   But I do have an issue when I try to cut it at the following times, and it's whether I try to cut the 1 minute one or the whole 2 min 13sec one.  If I cut with -ss 0 -t 1,   it produces a file that is 1s 22ms.  So that's not too bad that's fairly close to one second. 1.022 seconds is quite good for an attempt to cut one second.  If I cut with -ss 0 -t 2, it produces a file that is 3s 19ms   That is terrible! it's meant to be one second!  If I cut with -ss 0 -t 30 then it is fine it does 30s 1ms.  Or this video for example, also an mp4   C:\blah&gt;ffmpeg -ss 189 -i vid.mp4 -c copy -t 6 someoutput.mp4  C:\blah&gt;mediainfo someoutput.mp4   produces a duration of 8s 383ms  That is so wrong though 'cos it should be 6 seconds! So that is two seconds too long!  C:\blah&gt;ffmpeg -ss 191 -i vid.mp4 -c copy -t 6 someoutput2.mp4  C:\blah&gt;mediainfo output2.mp4   shows that ffmpeg produced a duration of 10s 403ms  which is crazily wrong 'cos the ffmpeg command specified 6 seconds!  So that's not just one second out, that is 4 seconds out!  That's almost 50% longer than it should be! Java reading a file into an ArrayList? What does it mean that a package is "set to manually installed?" When packages are already installed and I run an apt-get install &lt;package-name&gt;, sometimes it will print a line &lt;package-name&gt; is set to manually installed.  What does that mean? How to make a press proof with crop marks and color bars? A standard proof for press needs crop marks and color bars as below    With the package crop, the crop marks can be easily added.  But how about the color bar ?  Is there a way or a package to make it ?  An example is   \documentclass[b5paper,10pt]{article} \usepackage[utf8]{inputenc} \usepackage[cross,a4,center,info,axes]{crop}  \title{Test} \author{User} \begin{document} \maketitle \section{Fox} The quick brown fox jumps over a lazy dog. \end{document} There is already a similar post, that has an answer to my question, but it is quite short and I don't get it... So I am trying to show that  \begin{align} \{e_K |~ K \subset G~ conjugacy~ class \} \end{align} is a basis for $Z(\mathbb{Z}[G])$. Serre even writes "one immediately checks that the $e_K$ form a basis", but even reading the other post  I don't see that...  My approach to this is:  Be $x \in Z(\mathbb{Z}[G])$, which is equivalent to: $x = gxg^{-1}$ for all $g \in G$. Writing  $x = \sum\limits_{h \in G} \lambda_h h$, we get $x = \sum\limits_{h \in G} \lambda_h ghg^{-1}$. I don't really know how to go on at this point. I mean, I know that when $a,b \in K$ (same conjugacy class) you get $\sum\limits_{s \in K} \mu_s s = \mu\sum\limits_{s \in K} s$, since \begin{align} a = hbh^{-1} \Rightarrow \mu_a = \mu_b ~\forall a,b\in K \end{align} So all elements in the same conjugacy class have the same scalar.  But from here I don't really know how I can generate the $x = \sum\limits_{h \in G} \lambda_h ghg^{-1}$ with the $e_K$ ...  I would be really thankful, if someone could give me quite a basic proof that even I understand... Updating the SubscriberKey on the All Subscriber table through the API I'm trying to update an existing Subscriber in the All Subscriber Table through the API; i've set up the AppId in the App Center and have connected and created Subscribers, but i'm still having issues updating.  What call do I need to be making to do this? These are the ones i've been trying. Iridescence In Cycles? PHB says 'once per turn,' so, it would seem that a Rogue with Sentinel, for example, could "consistently" deal Sneak attack twice per round (whether the enemy triggers an OA or the melee weapon attack provided by the Feat).  So, can rogues deal sneak attack damage with any weapon attack outside their turn? (obv. one that meets the standard requirements--finesse, adv or adj ally) How to upgrade Git on Ubuntu Hardy? Standard Visitor visa refusal (Deception, V3.6(b)) and procedure for reapplying My mother acquired a criminal record in UK in 2010 when she was given a conditional discharge for 12 months and charged £85. This time while applying the on-line applications for her, I accidentally missed the criminal records column and replied "NO". The visa was refused with appeal and administrative review not allowed.  Can someone please suggest my next course of action for reapplying for a visa  It is an extremely urgent family matter, and I want the visa as early as possible. While researching I found in a blog some mention of a forwarding letter while reapplying.  Can someone please enlighten me on the entire issue? I'm using a PIN code to log on my computer. However, Num ⇩ is always turned off, even if on before shutting down my computer. This is quite annoying as I always have to retype my PIN.  I've tried the suggestions  (same as ). But with no result. If I don't enable num lock on log in screen and use the numbers above the normal keys, num lock turns on after log on. Then it's too late, unfortunately.  So, how can I make num lock enabled by default when logging in? Thanks.  Edit: While I still want to solve this, I've figured it is not such a big problem. I will need to press some key for the "enter PIN screen" to show, and using Num ⇩ works and enables num lock. But then, still annoying if the computer was only locked (with num lock on) and I turn it off, hehe. Convert all text from uppercase to lowercase and vice versa?