I'm trying to understand the behavior of $\sum_{n=1}^\infty \frac{1}{n} z^{n!}$ on the unit circle.  Since for each $m$th root of unity $\zeta_m$ $$\sum_{n=1}^\infty \frac{1}{n} \zeta_m^{n!} = C + \sum_{n=m}^\infty \frac{1}{n} = \infty$$ holds for some $C \in \mathbb{C}$, the series diverges for all $e^{\varphi \pi i}$ with $\varphi \in \mathbb{Q}$.  But what happens for $\varphi \in \mathbb{R} \setminus \mathbb{Q}$?   Does the series diverge everywhere, or are there points where it is convergent?
I recall from complex analysis a while ago that the series $$\sum_{n=1}^\infty \frac{z^{n!}}{n}$$ has radius of convergence 1, diverges on roots of unity, and converges elsewhere on the circle. It's simple enough to see that it has to diverge on roots of unity: consider $z$ such that $z^k = 1$ for some $k \in \mathbb{N}.$ Plug this into the series and you see that past the $k$'th term, $z^{n!} = (z^k)^m = 1$ for some $m \in \mathbb{N}$; hence, the series becomes harmonic and must diverge.  But I don't recall how to show that the series converges for other values on the circle. If this is simple I'd like just a hint, if it is involved I would prefer a sparse outline of the proof. Thank you!
Is the Jekyll blog dead? I noticed that the  is apparently abandoned and the  is back to WordPress.  I'd be interested to know what were the decisions that led to switching back. Was the content creation process hindered by Markdown + git? Were there growing pains that couldn't be solved? Computer unstable at lower temperatures My computer is playing some strange joke at me, and I'm at a loss.  Everybody knows that hot temperatures causes many components to become unstable and even cause damage. When I built this machine, I was careful to do a bunch of experiments with different airflow configurations, with CPU-and-GPU intensive benchmarks running, and measuring the temperature at different points to find the best configuration.  Generally, the internal temperature of this machine is very close to ambient temperature (with 2 exceptions: the GPU always runs ~10°C above ambient, and the CPU goes to at most ~12°C above ambient when running CPU-intensive applications).  Now comes the problem: at really cold mornings. At cold mornings (~10°C (50°F) or below), the system shows really strange general instability. Beeps from the BIOS, the operating system displays strange messages about memory corruption and PCI devices that failed to initialize, and then, everything hangs. The computer hangs to the point that all USB devices turn of. Hitting the Reset button blanks the screen but the mainboard fails to proceed to the boot.  I never thought that low temperatures could cause computer instability. And 10°C is not that cold, it is still within operating temperatures of all components as far as I checked.  I'm quite sure it is related with the temperature. There were too many cold and warm mornings that demonstrated that the problem only showed in the cold ones. I'm working around this problem by letting the computer sit at the GRUB menu for some 5-10 minutes before letting the OS start, or blowing hot air inside the case if I'm in a hurry.  I don't know what should I approach next to find the source of this problem. Since it is difficult to reproduce (I need a cold morning, and the computer must be cooled down (powered off for many hours)), I need to plan ahead the experiment, since it stabilizes after a few minutes on.  Machine is an ASUS P5VDC-X equipped with Intel P4 64-bit 3GHz, 2GB of DDR2 RAM, a 500W PSU and an nVidia GeForce 9600GT. Everything else was already disconnected/removed before and the problem persisted, so the problem should be in one of these remaining parts.  Any help is appreciated! Recolor Artwork: single swatch to multiple swatches I have an Illustrator document with a lot of objects. All the objects are black, same swatch.  I have a swatch group with a few dozen coloured swatches. I wish to apply these swatches to all the many black objects. It doesn’t have to be in any particular order, but each object should be coloured individually—I don’t just want to change black into some other colour, but to dozens of other colours. What I really want to avoid is having to select each object individually and then applying a different swatch to each.  I’ve never quite managed to get my head properly around the Recolor Artwork feature in Illustrator, and I cannot for the life of me get it to do anything like this. The “New” box in the Recolor Artwork dialogue box appears to accept only one colour.  Is there some way to do this using the Recolor Artwork feature? Or using some other feature that I’m not aware of? Bonus points if there’s a way to control the order in which the colours are applied; e.g., specifying an order and then click on each object in turn to apply each swatch being ‘iterated’ through in that order.  (I would prefer to avoid having to script it, but I will if there’s no other way.) Suppose that   $$f(x)=1+\sum_{n=1}^\infty a_n \frac{x^n}{n!}\ \ \forall \ x\in \mathbb{R}$$   where $\sup_{x&gt;0}\left|e^xf(x)\right| &lt; \infty$ and $\sup_{n\in\mathbb{N}} |a_n|&lt; \infty$.      Prove that $a_n = (-1)^n$ , $\forall n\in \mathbb{N}$   It seems amazing to me. What we need to prove is $f(x)=e^{-x}$. It seems insufficient to prove this strong conclusion, but actually it is true and all the "counterexamples" I found were wrong.  My attempt  Put $g(x)=e^x f(x)$. $$\left|g^{(n)}(0)\right|=\left|\sum_{k=0}^n \binom{n}{k} f^{(k)}(0)\right|\le 2^n\sup_{n\in\mathbb{N}} |a_n|  $$ Put $h(x)=g(\frac{x}{2})$. Thus  $$h^{(n)}(0)=\frac{1}{2^n}g(0) \le \sup_{n\in\mathbb{N}} |a_n|$$ which implies that $$h(x)=1+\sum_{n=1}^\infty b_n \frac{x^n}{n!}\ \ \forall \ x\in \mathbb{R}$$ where $$|b_n|\le \sup_{n\in\mathbb{N}} |a_n| \ \ \forall \ n\in\mathbb{N}  \,\,\,\,\,\&amp; \,\,\,\,\,  \sup_{x&gt;0}\left|h(x)\right| &lt; \infty $$   And hence if $b_k&lt;0$, then there exists $l&gt;k$ such that $b_l&gt;0$. I want to yield a contradiction by supposing this, but I failed.  Any hints or other new ideas? Thanks in advance!   (I heard that this problem can be solved by complex analysis. This is the reason why I attach the complex-analysis tag.) I tried:   Launching Ubuntu Software Centre I typed in "air" and found "adobe air" It says "available from the 'maverick-partner' source. and has a button "use this source" I clicked "use this source", I saw a "in progress (1)" icon come up in the side bar, and disappear, but nothing happened/changed.   I'm stuck.  I have actually already added the maverick-partner source to apt so I could install sun-java6-jdk. Fortunately, I can still login with this email address that I have lost access to.  I have been searching around here for a way to update it to the current one I use.  I have found many answers.  But they all give instructions to click on something not displayed or in one case, a page that prompts me to login via another email I no longer have the password for. I need current up to date instructions on how to change my email address or add another email-based login for this same user name that I am posting this with. I want to set an environment variable from the command prompt and then be able to access it globally (for instance, I should see it by going to System -> Environment Variables).    When I use the set command, it isn't accessible in a new cmd session.  set NEWVAR=SOMETHING echo %NEWVAR%   Related questions: Is there a word for someone that always has to be right? 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Who's at fault with this error concerning csquotes and LuaLaTeX? I ran into this with the package lilyglyphs but I'm not sure whose fault this error is.  When I compile the following with lualatex:  \documentclass{article} \usepackage{csquotes} \MakeOuterQuote{"} \EnableQuotes \begin{document} \directlua{print("hello")} \end{document}   I get an error saying   \directlua]:1: unexpected symbol near '\'. l.6 \directlua{print("hello")}    Now suppose that I do not directly write \directlua in my document, but I include a package that does it internally. Such as   \documentclass{article} \usepackage{fontspec, csquotes, lilyglyphs} \MakeOuterQuote{"} \EnableQuotes \begin{document} \wholeNote \end{document}   I have no knowledge why compilation fails, because the error now becomes   ! Missing number, treated as zero. &lt;to be read again&gt; \begingroup l.7 \end       {document}    From my user point of view I did nothing wrong, because I'm using the public API provided by the packages. The csquotes documentation specifically says:     Note that all characters are automatically checked for validity as they are allocated. This package will reject characters which are unsuitable as active quotes... In sum, the following   characters will be considered as reserved by this package:   A–Z a–z 0–9 . , ; : ! ? ' - # $ % &amp; ^ _ ` ~ \ @ * { } [ ]    So my question is whose fault is this error?   My fault, the user's fault, because I should have known (but how?) that even though the quote is not listed as unsuitable in the csquotes manual, it actually is? The fault of the implementation or documentation of csquotes for failing to mention " as unsuitable? The fault of the implementation of the \directlua command because it doesn't prevent LaTeX catcode changes/active characters from impacting its argument? A shortcoming of TeX itself because of its global state based architecture? Command Options on Command Line - Details and Functions? 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