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Let $f$ be a real, continuous function defined on $[0,\infty)$ such that $xf(x)$ is increasing for all sufficiently large values of $x$. Show that if   $$\int_0^x f(y)\,dy \sim Ax^\alpha \quad \left(\,x\to \infty\right)$$  for some positive constants $A$ and $\alpha$, then  $$f(x)\sim \alpha Ax^{\alpha -1} \quad \left(\,x\to \infty\right).$$  Clearly, I have to use differentiation somewhere, but I don't know how to manipulate $\lim_{x\to \infty}\frac{\int_0^x f(y)\,dy}{Ax^\alpha}=1$ to get the desired result.  From suggestions given, I know that by L'hospital's rule, it's enough to show that the limit $\lim \frac{f(x)}{\alpha Ax^{\alpha -1}}$ exists, and this limit equals $ \lim \frac{xf(x)}{\alpha Ax^{\alpha}}$. From here, I'll need to use the given assumption that $xf(x)$ is eventually increasing. But how can I show the existence of the limit based on these facts?  I would greatly appreciate any solutions, hints or suggestions. Proving $\sum_{k=1}^n k k!=(n+1)!-1$ Parent chat user still links to discuss.area51 When viewing a user who has their parent user on chat set to area51, it still links to discuss.area51, instead of the new URL, area51.meta. See:      So can we update these links? Calculating de Broglie wavelength Is there a way to make an Android phone (HTC Desire HD) act as Bluetooth speakers or headset?  I'd like to use the phone as wireless headphones whilst watching a movie on a computer (running Ubuntu). Here's my source:  #!/bin/bash  echo "Running script to free general cached memory!" echo ""; echo "Script must be run as root!"; echo ""; echo "Clearing swap!"; swapoff -a &amp;&amp; swapon -a; echo ""; echo "Clear inodes and page file!"; echo 1 &gt; /proc/sys/vm/drop_caches; echo "";   It clears caches and stuff, and it echoes that it needs to be run as root in the terminal. I basically just want the script to cease running if it detects it's not being executed as root.   Example:  "Running script to free general cached memory!" "Warning: script must be run as root or with elevated privileges!" "Error: script not running as root or with sudo! Exiting..."   If run with elevated privileges, it just runs as normal. Any ideas? Thanks! Differentiation - simple case In the book calculus made easy, page 22 the case of the negative power for   $y=x^{-2}$    $$\begin{align}  y+dy &amp; =(x+dx)^{-2}\tag{1}\\ \\ &amp; = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} \end{align}$$  How do they get from $(1)$ to $(2)$?  And why isn't the binomial function applied to the second line but applied on the third line (the line that comes after line $(2))$? My advisor wants me to include a name of someone who has no contribution to the paper My first advisor wants me to include my second advisor's name in a paper that I worked on before she (2nd advisor) came along.   Should I include her name even though she didn't do anything in that paper? I don't think she even read it. I am using Sql Server 2008 on Small Business Server 2003; client is using WinXP.   I added a user to my Active Directory Security Group; Why can't this user immediately access the database? It seems there is a delay before the user is recognized in Sql Server.   I am using AD Security Groups for permissions expressly so that I don't need to add individual users in Sql Server. So effectively I don't need to do anything but add the user to the AD Security Group in order to grant access.  But for some reason Sql Server doesn't immediately recognize the addition. I've seen this a number of times. I add the user to the group but that user can't access data until the next day. It seems that it doesn't query the Active Directory in real time. Can you confirm that is so?  What do I need to do so that Sql Server "refreshes" the list of users from Active Directory? What is dependency injection? Did spacetime start with the Big Bang?  I mean, was there any presence of this spacetime we are experiencing now before big bang? And could there be a presence/existence of any other space-time before the big bang? How does this JavaScript/jQuery syntax work: (function( window, undefined ) { })(window)? Intersection of powers of an ideal in a Noetherian ring Given a Noetherian ring $R$ and a proper ideal $I$ of it.  Is it true that $$\bigcap_{n\ge 1} I^n=0$$ as $n$ varies over all natural numbers? If not, is it true if $I$ is a maximal ideal? If not, is it true if $I$ is the maximal ideal of a local ring $R$? If not, is it true under additional assumptions on $R$ (like $R$ is regular)? Creating polygon from hole in QGIS? What level should I be to kill Alduin I am level 31 and I have the Elder Scroll. I am not sure if my level is high enough to go to Whiterun and trap one of Alduin's minions. What level should I be before I attempt it? Proving convergence of sums: If $\sum_{n=1}^{\infty}a^2_n$ and $\sum_{n=1}^{\infty}b^2_n$ converge then $\sum_{n=1}^{\infty}a_nb_n$ converges Automatic vacuum fails with `ERROR: XX000: cache lookup failed for type 0` I've been noticing something like this popping up every 15 seconds in my server logs:  2020-01-30 21:10:30 UTC::@:[24969]:ERROR: XX000: cache lookup failed for type 0 2020-01-30 21:10:30 UTC::@:[24969]:CONTEXT: automatic vacuum of table "myschema.mytable" 2020-01-30 21:10:30 UTC::@:[24969]:LOCATION: get_typlenbyval, lsyscache.c:2036   When I manually run a VACUUM ANALYZE myschema.mytable; it runs without any error, and the error in my logs goes away. However, inevitably, it returns again at some point.   The table in question is defined as follows:  tfs_dev=&gt; \d myschema.mytable;                                                    Table "myschema.mytable"    Column    |   Type    | Collation | Nullable |                                  Default -------------+-----------+-----------+----------+---------------------------------------------------------------------------  level       | text      |           | not null |  id          | integer   |           | not null | nextval('myschema.mytable_id_seq'::regclass)  text_id     | text      |           |          |  name        | text      |           |          |  geom        | geometry  |           |          |  valid_dates | tstzrange |           |          | tstzrange(NULL::timestamp with time zone, NULL::timestamp with time zone)  adjacent    | integer[] |           |          |  valence     | integer   |           |          |  color       | smallint  |           |          | 0 Indexes:     "mytable_pkey" PRIMARY KEY, btree (level, id)     "mytable_text_id_key" UNIQUE CONSTRAINT, btree (text_id)     "mytable_gix" gist (level, geom, id, text_id, name)     "mytable_spgix" spgist (geom)     "mytable_text_id_ix" btree (text_id, level, id, name) Referenced by:     TABLE "otherschema.othertable" CONSTRAINT "othertable_mytable_fk" FOREIGN KEY (level, level_id) REFERENCES myschema.mytable(level, id)     TABLE "otherschema.othertable2" CONSTRAINT "othertable2_mytable_fk" FOREIGN KEY (level, level_id) REFERENCES myschema.mytable(level, id)   Many other posts with a similar error seem to be about a foreign data wrapper, which is present in this database, but not with this table/schema. Increase the highlighted button bezel thickness (GNOME) I had always kind of wondered why we didn't see interference in things like windows -- we were taught that the condition is that the thickness of the film/slab/medium just has to be an integral number of wavelengths, $d=m\lambda$. My logic was, if visible light has wavelengths ~400-780nm, any window pane has got to be an integral number of wavelengths (or close to integral) for some of the wavelengths in the visible.  For example, if a window is, $6mm = 6\times10^6nm$ thick, then you should see interference for wavelengths (and some range around them) 200nm, 250nm, 300nm, 400nm, 500nm, 600nm... Because 6mm is an integral number of each of those.  Obviously that would be really annoying and isn't the case. I found  (warning: PDF) source that seems to explain it and I was wondering if what they're saying is sound. On the 10th page (p171 for them), it says that the reason we don't see this interference is that no window is that smooth, it has a variance in the thickness of at least several wavelengths, so the interference effect is effectively average out among all those thicknesses, which usually gets rid of it. They do the math and show that.  Is that the reason?  If so, that leads me to two questions: Could we, if we were really careful and used something like MBE, make a glass window with that sort of accuracy, and then see interference effects in the optical range? I know that quartz crystal monitors can easily give you nanometer accuracy for evaporators and sputterers, but they also have pretty limited growth rates.  My second question is, then, do we see interference effects in glass windows with electromagnetic radiation of a comparable wavelength, like microwaves?  Thanks! $\lim_{x \rightarrow \infty} \lim_{n \rightarrow \infty} f_{n}(x) = \lim_{n \rightarrow \infty}\lim_{x \rightarrow \infty}f_{n}(x) $ in case of uniform convergence of $f_{n}(x)$ to $f(x)$  I used this while I was given that $f_{n}(x)$ converges to $f(x)$ uniformly. Is this correct and I was also thinking to prove this!