Wolframalpha tells that $$\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}e^n$$ diverges since $$\lim_{n\to\infty}(\frac{n}{n+1})^{n^2}e^n=\sqrt{e}$$. How do you calculate this limit?   And also it tells that $$\sum_{n=0}^\infty (-1)^n(\frac{n}{n+1})^{n^2}e^n$$ is not convergent, but it didn't specify the reason. Can you give me one?
I want to find the interval of convergence of $$\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(2x)^n$$ By the root test, $\sqrt[n]{|(\frac{n}{n+1})^{n^2}(2x)^n|}=|(\frac{n}{n+1})^{n}(2x)|$  And $(\frac{n}{n+1})^n=(1-\frac{1}{n+1})^n\to\frac1e$ as $n\to\infty$  So I've got an open interval(of convergence) s.t. $|\frac1e 2x|&lt;1\implies |x|&lt;\frac e2$.  Now I need to figure out whether the series is convergent or not at the end points, $x=\frac e2, -\frac e2$.  So there are two series I need to deal with, namely, $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}e^n$ and $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(-1)^ne^n$.  Suppose $(\frac{n}{n+1})^{n^2}$ is increasing. Then $$(\frac{n}{n+1})^{n^2}&lt;(\frac{n+1}{n+2})^{(n+1)^2}\iff 1&lt;(\frac{n+1}{n+2})^{(n+1)^2}(\frac{n+1}{n})^{n^2}$$ $$\iff 1&lt;(1-\frac{1}{n+2})^{2n+1}(1+\frac{1}{n^2+2n})^{n^2}$$  But $(1-\frac{1}{n+2})^{2n+1}\to\frac{1}{e^2}$ and $(1+\frac{1}{n^2+2n})^{n^2}&lt;(1+\frac{1}{n^2})^{n^2}\to e$  So I have $1&lt;\frac1e&lt;1$, which is a contradiction. Hence $(\frac{n}{n+1})^{n^2}$ is not increasing. But I don't know if it's decreasing. (is it true that $(1+\frac{1}{n^2+2n})^{n^2}\to e$?)  If $(1+\frac{1}{n^2+2n})^{n^2}\to e$ is true, then $(\frac{n}{n+1})^{n^2}e^n$ converges to some positive number, so $\sum(\frac{n}{n+1})^{n^2}e^n$ diverges. But I don't know if it's decreasing: can't apply the alternating series convergence test(Leibniz's criterion).  So, can you help me with the two end points?
A and B disjoint, A compact, and B closed implies there is positive distance between both sets Is propagation of uncertainties linear? How can gravity affect light? I understand that a black hole bends the fabric of space time to a point that no object can escape.   I understand that light travels in a straight line along spacetime unless distorted by gravity. If spacetime is being curved by gravity then light should follow that bend in spacetime.  In Newton's Law of Universal Gravitation, the mass of both objects must be entered, but photon has no mass, why should a massless photon  be affected by gravity in by Newton's equations?  What am I missing? How can I play Pokémon GO without Internet? I downloaded Pokémon GO on my Samsung Galaxy Express and I can play it at home, but when I go outside I can't play because there is no Internet.  How can I play Pokémon GO without Internet outside? Difference between "been to" and "been in" Can Git hook scripts be managed along with the repository? We'd like to make a few basic hook scripts that we can all share -- for things like pre-formatting commit messages.  Git has hook scripts for that that are normally stored under &lt;project&gt;/.git/hooks/.  However, those scripts are not propagated when people do a clone and they are not version controlled.    Is there a good way to help everyone get the right hook scripts?  Can I just make those hook scripts point to version controlled scripts in my repo? User with low reputation shows as having modified thread How can this user , reputation 3, show in  as having "modified 7 hours ago", but on the modified thread, , not show at all?  This user also has two medals, which is somewhat puzzling. It has come up in game a couple of times that a player might want to cast a spell that has a casting time of 1 bonus action using their "main" action (if they have another bonus action they also want to take on that turn, such as giving bardic inspiration, or controlling a Bigby's hand, etc.)  On the face of it, it seems obvious that something (a bonus action) that is usually much faster than a full action could be done as your full action.  Although the question comes up most often with respect to spellcasting, if I house rule this, I would rule that any bonus action can be taken as a regular action instead; however, I would not allow the same type of bonus action to be taken twice (so no giving bardic inspiration to two allies on the same turn, for instance).   Are there any abusive or overpowered combinations I should be wary of if I were to allow a character to take 2 bonus actions instead of one regular action and one bonus action on a turn?   The issue of casting two bonus-action spells would not come up because the rule against casting 2 spells on your turn unless one of them is a cantrip with a casting time of 1 action would still be in effect:  PHB p. 203 (under Bonus Action casting time)     You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.   I know it's hard to prove/justify a negative answer to a question like this, but I'd be happy to get answers that say you don't think there would be any issues if you describe how you came to that conclusion. This is a question from my proofs course review list that I have had trouble understanding.   I understand the concept of disjoint sets.  I'm not sure what they mean by countable. How would one prove a set is countable and furthermore, that the union of two countable sets is countable.  Please help me get started with the question by understanding this concept. When you're logged in to a Stack Exchange account, it's possible to add and remove logins (associated OpenID/OAuth accounts) without performing any form of reauthentication.  This is not a full vulnerability in itself, but it provides an extremely easy way for an attacker who's found a temporary XSS vulnerability to turn that into a permanent account hijack. An XSS exploit on its own may only provide a minute of indirect access to a user's account. But with this login configuration, the attacker could use that minute to automatically:   Add a new attacker-controlled login to the account. For example:   The XSS JavaScript would generate a random identifier, and use that to construct a login URL of the form http://hacker.example/openid/1234567. The hacker.example server would automatically authenticate the first OpenID login request made with a given identifier, allowing it to be added to the account without user interaction. All subsequent login attempts would be automatically rejected for the user. The hacker.example server would notify the operator with the account name and the generated identifier, and would allow them to login to it.  Remove all existing logins from the account. Log the user out of their account, with no way to get back in.   If this exploit were used to hijack moderator and staff accounts it could makes it more difficult to quickly respond to an attack, creating a greater opportunity for extended abuse.  This is a broader instance of the same fundamental vulnerability which was previously-reported as . That instance is a little more straightforward to fix: just ask for the password again when changing it. However, safe reauthentication with OpenID/OAuth logins is more complicated. Most users' OpenID providers will be configured to automatically re-authenticate upon request, so there would be minimal security gain from requesting that.  Instead or as well, my suggestion is that for users that have a verified email address, require an emailed confirmation link to be clicked before committing any login changes.    I'm not a real security expert, but I've just found that fellow Stack Overflow user , who is, previously described this vulnerability in his 2012 blog post :     If an attacker can add additional login accounts then they can create a persistent mechanism to get back into that Stack Overflow account at any time without needing to hijack the session. Now of course the victim can always go into their account settings and identify the presence of the new login, but it’s a question of how much damage is done before then.      How would an attacker mitigate the risk of having their newly added account removed? Just remove the victim’s original account(s) instead.      [...]      Finally, this is also a lesson for those considering using third party authentication providers. Yes, it’s great that you can farm off this work to someone who has a solid implementation, but you still have responsibility for securing the authenticated session on your site. If you choose not to, be aware of the risks because certainly we’ve seen them exploited before. Remove votes to close question X as a duplicate of Y if question Y is closed as a duplicate of X I just happened across , which has one close vote suggesting that it be closed as an exact duplicate of .  This suggestion is perfectly correct: these are exactly identical questions asked by the same user.  However, there's not much point in closing question A as a duplicate of B, since question B has already been closed as a duplicate of A.  Yet the close vote remains, doing nothing but wasting reviewers' time.  This is not the first time I've seen this happen, so I'd like to suggest that the Stack Exchange software should do something about it.  In particular, I'd like to suggest the following feature:     When a question Y is closed as an exact duplicate of another question X, any votes or flags to close X as a duplicate of Y should be automatically discarded.   Actually, I'm kind of surprised that the SE software doesn't do that already.  If anyone else thinks that this constitutes an actual bug, feel free to retag it. Don't impose delay loading Close dialog when you haven't just loaded it I am to prove by induction that given that $a+1/a$ is an integer (i.e. belongs to Z  ) then $a^n+1/a^n$ is an integer too. I'm pretty much clueless here. Thanks in advance. How can I group numerical data into naturally forming "brackets"? (e.g. income) The following describes what I'm trying to accomplish, but it's possible an alternative problem statement can describe my goal:  I want to    divide the following numbers into groups where the variances of the numbers within each group are not too large, and the differences between the averages of groups are not too small  compare the distribution obtained in the end with the "perfect" ones and see how "different" it is from being perfect.     Layman's explanation of goal  , and determine the "income brackets" each population is in.  The income bracket is supposed to be self-adjusting based on the input data.  My goal is to ultimately measure or calculate the difference between the income brackets.  I assume there will be many brackets, and want to see how far "apart" each tier is.  Here is a sample of hourly income for a sample set of a population of 20, and a total income of 3587:  Population= 10                   pop=2   population=5              population =3 10, 11,13,14,14,14,14,14,15,20,  40,50  ,90,91,92,93,94      999,999,900    How can I use mathematical concepts to group, sort and analyze data that acts like income distribution over a given population?  At the end of the calculation, I want to determine tiered income distribution, where a perfect distribution would look (something) like this   (each person makes $10 more per hour than the previous; total is 3587) 89, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 209, 219, 229, 239, 249, 259, 269, 279   or this:   (evenly distributed groups of people make the same per hour)  (gaps between income groups is consistent and not "too far") (income total is 3587) 99 99 99   129 129 129   159 159 159   199 199 199  229 229 229  269 269 269    Question  How should I analyze the population groups, and measure the gap in a way that will tell me how much is needed to make it more like the last two model sets listed above? How to shuffle all songs in iOS 8.4 music app In the old version of the music app you could shuffle all songs when songs were selected:      This feature seems to be absent in iOS 8.4 and the new music app:      What are the options for shuffling all my songs in the new version of the music app under iOS 8.4? If $\gcd (a,b) = 1$, what can be said about $\gcd (a+b,a-b)$? Prevent Use of the Back Button (in IE) how do i change the color of typeface on photoshop? centering column headers in siunitx table column table of contents: adding a vertical space In a table of contents that has been generated with LaTeX's \tableofcontents command, I would like to add a vertical space before a Chapter item in the table of contents. One way that I have in my notes is to use  \addtocontents{toc}{\protect\vspace{10pt}}   at a desired location in the document so that this gets processed by \tableofcontents to include an appropriate vertical space in the table of contents. I have two questions:   Is the above the correct way to get a vertical space in a table of contents? Why is it necessary to use the \protect command here?   Thank you...