I am interested in the limit     $$ \lim_{n \rightarrow \infty} \sqrt[n]{n^3}$$   Can we simply conclude that: $$ \lim_{n \rightarrow \infty} (\sqrt[n]{n})^3= 1^3=1.$$ I have proven that $\sqrt[n]{n}\rightarrow1$ earlier in this textbook. Also since the limit of a power is the power of the limit.
Thomson et al. provide a proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$ in . It has to do with using an inequality that relies on the binomial theorem:  I have an alternative proof that I know (from elsewhere) as follows.  Proof. \begin{align} \lim_{n\rightarrow \infty} \frac{ \log n}{n} = 0 \end{align} Then using this, I can instead prove: \begin{align} \lim_{n\rightarrow \infty} \sqrt[n]{n} &amp;= \lim_{n\rightarrow \infty} \exp{\frac{ \log n}{n}} \newline &amp; = \exp{0} \newline &amp; = 1 \end{align}  On the one hand, it seems like a valid proof to me. On the other hand, I know I should be careful with infinite sequences. The step I'm most unsure of is: \begin{align} \lim_{n\rightarrow \infty} \sqrt[n]{n} = \lim_{n\rightarrow \infty} \exp{\frac{ \log n}{n}} \end{align} I know such an identity would hold for bounded $n$ but I'm not sure I can use this identity when $n\rightarrow \infty$. Question: If I am correct, then would there be any cases where I would be wrong? Specifically, given any sequence $x_n$, can I always assume: \begin{align} \lim_{n\rightarrow \infty} x_n = \lim_{n\rightarrow \infty} \exp(\log x_n) \end{align} Or are there sequences that invalidate that identity?  (Edited to expand the last question) given any sequence $x_n$, can I always assume: \begin{align} \lim_{n\rightarrow \infty} x_n &amp;=  \exp(\log \lim_{n\rightarrow \infty} x_n) \newline &amp;=  \exp(\lim_{n\rightarrow \infty} \log x_n) \newline &amp;=  \lim_{n\rightarrow \infty} \exp( \log x_n) \end{align} Or are there sequences that invalidate any of the above identities? (Edited to repurpose this question). Please also feel free to add different proofs of $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$.
Basic Excel currency format with Apache POI I'm able to get cells to format as Dates, but I've been unable to get cells to format as currency... Anyone have an example of how to create a style to get this to work?  My code below show the styles I'm creating... the styleDateFormat works like a champ while styleCurrencyFormat has no affect on the cell.  private HSSFWorkbook wb; private HSSFCellStyle styleDateFormat = null; private HSSFCellStyle styleCurrencyFormat = null;   ......  public CouponicsReportBean(){     wb = new HSSFWorkbook();     InitializeFonts();  }  public void InitializeFonts() {     styleDateFormat = wb.createCellStyle();     styleDateFormat.setDataFormat(HSSFDataFormat.getBuiltinFormat("m/d/yy"));       styleCurrencyFormat = wb.createCellStyle();     styleCurrencyFormat.setDataFormat(HSSFDataFormat.getBuiltinFormat("$#,##0.00"));  } Is there any evolutionary/physiological advantage to multi-lateral symmetry? Just wondering ; I get the impression a large number of multicellular organisms are multi-lateral in their physical structure. Why would evolution/mutation have retained multi-lateral symmetry in an organism? What advantage/s does symmetry provide an organism?  EDIT:  By multi-lateral symmetry I mean at-least bi-lateral symmetry (as in the case of many beings such as elephants, crabs and fishes), but perhaps more than bi-lateral too (starfish come to mind)  END EDIT: Is it possible to annotate equations (i.e. arrow signs/curly braces) to define each variable? I have quite a simple question. I was wondering if it was possible to do something like this in LaTeX:    But formatted a little more nicely.  It doesn't need to be arrows necessarily. Just some form of annotation marker. How do I draw text at an angle using python's PIL? Using Python I want to be able to draw text at different angles using PIL.  For example, imagine you were drawing the number around the face of a clock.  The number 3 would appear as expected whereas 12 would we drawn rotated counter-clockwise 90 degrees.  Therefore, I need to be able to draw many different strings at many different angles. A good reference on optimal control theory Ok, so I am reading about decision making and I came across this subject. Fortunately it has a  , but the point is I want to see some examples, and learn to solve regular problems of this field. Of course I can go for the book by Pontryagin but then I don't think that would be an educative book. Does anyone has a suggestion? I would be happy if it also covers LQ problems. P.S. I studied math, therefore I appreciate a book with rigorous approach to the subject. What's the Best Way to Shuffle an NSMutableArray? If you have an NSMutableArray, how do you shuffle the elements randomly?  (I have my own answer for this, which is posted below, but I'm new to Cocoa and I'm interested to know if there is a better way.)    Update: As noted by @Mukesh, as of iOS 10+ and macOS 10.12+, there is an -[NSMutableArray shuffledArray] method that can be used to shuffle. See  for details. (But note that this creates a new array, rather than shuffling the elements in place.) How do you solve a recurrence with a summation function inside: $t(n) = 1 + \sum\limits_{j=0}^{n-1} t(j)$ Show that  $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$  is the same as  $$t(n) = 2^n$$  Initial condition $t(0) = 1$ Let $z_1, z_2, z_3,$ ...be a sequence of independent random variables s.t. $P(z_i=1)=P(z_i=-1)=\frac 1 2$. Does this sequence converge almost surely?  My intuition is telling me that the sequence will approach the random variable Z? i.e. $z_n-&gt;Z$ as $n-&gt;\infty$ and Z is just another RV s.t. $P(Z=1)=P(Z=-1)=\frac 1 2$.  Here is my confusion, if a sequence converges almost surely, then $z_i$ should be different for each term s.t. it approaches Z in the end; but in this sequence $z-i$ is the same for every term. So I dont udnersntand. What's the term for denying something so as to raise doubt? Topical example:  'There was a scuffle outside the local pub. There appears to be no connection with religious extremists.'  ...but now you're wondering if there was. It's a nasty tactic used to raise fear and doubt - deliberately calling attention to something by explicitly denying it.  What's the term for this? Splitting pieces of a mesh into a new object Reducing size of KML file (after converting from Shapefile) The command history between sessions is not getting saved. I'm using guake and the history for the session is working fine. I noticed that .bash_history had some commands I executed in sudo -s mode and tried the same again and all the commands while in the session got saved so I tried: chmod 777 .bash_history  Now the old commands appear at the start of a session but no new commands are getting saved. I need to have a list of all available view modes for an entity type so I can use it as the #options in a select field.  Any ideas? Thanks How to completely disable the Windows taskbar? I've installed Taskbar eliminator, while it works most of the time, alerts being the taskbar back.  I've also installed object dock which can disable the taskbar, object dock appears during window changes.   Has anyone had any luck permanently removing, hiding the taskbar in Windows 7?  Update  I found an application that seems to be working here:    Second update  I took the leap and installed a shell replacement (). So far so good. What does ‘Sport’ mean when you say ‘the new Apple iPad sports cameras for video conferencing’? Can you prove that if $p$ is a prime greater than $5$, then the sum of the squares of the quadratic nonresidues modulo $p$ is divisible by $p$?  Note that I have just proved that the sum of the quadratic residues modulo $p$ is divisible by $p$ for $p$ greater than $3$. How can I reuse a Common Table Expression Is there a mass spectrometer software that can measure the mass of a proton Shell's process group ID = foreground job's process group ID? I decisively conclude that Shell's process group ID = foreground job process group ID from the 2 sources below. When a background job is selected to run in the foreground, is it the shell's process group ID that changes to the foreground job's process group ID or the reverse?  1.     To facilitate the implementation of the user interface to job control, the operating system maintains the notion of a current terminal process group ID. Members of this process group (processes whose process group ID is equal to the current terminal process group ID) receive keyboard-generated signals such as SIGINT. These processes are said to be in the foreground. Background processes are those whose process group ID differs from the terminal’s; such processes are immune to keyboard-generated signals.    2.        Addition:  $sleep 3000 &amp; $sleep 2000 &amp; $ps xao pid,ppid,pgid,sid,tty,comm | grep tty  PID     PPID    PGID    SID     TTY     COMMAND 1153    1135    1153    1153    tty1    bash 1173    1153    1173    1153    tty1    sleep 1189    1153    1189    1153    tty1    sleep 1219    1153    1219    1153    tty1    ps 1220    1153    1219    1153    tty1    grep Cutting a circular hole in a hollow cylinder without distorting the cylinder Whenever I try to cut out a circular hole in either a cylinder or sphere the polygons around the circular hole always have distorted normals after smoothing. Is there anyway to fix this. I've tried using the remesh modifier but wind up with some pretty ugly results.