$f(x)= \Sigma_{n=1}^{\infty} \frac {\sin nx^2}{1+n^3}$, $x\in {\mathbb{R}}$  I can show continuity by using uniform convergence and the Weierstrass M test. But I don't know how to do differentiability. Any help would be highly appreciated. Thank you.
Show that the following series of function defines a continuous differentiable function function in $\mathbb R$. $$\sum_{n=1}^{\infty}\frac{\sin (nx^2)}{1+n^3}.$$  We have , $|f_n(x)|=\left|\frac{\sin (nx^2)}{1+n^3}\right|\le \frac{1}{1+n^3}\le \frac{1}{n^3}=M_n \text{ (say) }$  As, $\sum M_n$ is convergent so, the given series is uniformly convergent. Also , each $f_n(x)$ is a continuous function in $\mathbb R$. So, the given series converges to a continuous function , say $f(x)$.  But how we can show that $f(x)$ is differentiable function ?  Please help...
To what extent is WhatsApp's statement on secure messaging realistic? Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as well? Can Windows use a Linux shell instead of cmd? How to interpret autocorrelation plot in MCMC I am getting familiar with Bayesian statistics by reading the book Doing Bayesian Data Analysis, by John K. Kruschke also known as the "puppy book". In chapter 9, hierarchical models are introduced with this simple example: \begin{align} y_{ji}   &amp;\sim {\rm Bernoulli}(\theta_j)  \\ \theta_j &amp;\sim {\rm Beta}(\mu\kappa, (1-\mu)\kappa)  \\ \mu      &amp;\sim {\rm Beta}(A_\mu, B_\mu)   \\ \kappa   &amp;\sim {\rm Gamma}(S_\kappa, R_\kappa) \end{align} and the Bernoulli observations are 3 coins, each 10 flips. One shows 9 heads, the other 5 heads and the other 1 head.  I have used pymc to infer the hyperparamteres.  with pm.Model() as model: # define the          mu = pm.Beta('mu', 2, 2)     kappa = pm.Gamma('kappa', 1, 0.1)     # define the prior     theta = pm.Beta('theta', mu * kappa, (1 - mu) * kappa, shape=len(N))     # define the likelihood     y = pm.Bernoulli('y', p=theta[coin], observed=y)      # Generate a MCMC chain     step = pm.Metropolis()     trace = pm.sample(5000, step, progressbar=True)     trace = pm.sample(5000, step, progressbar=True)   burnin = 2000  # posterior samples to discard thin = 10  # thinning  pm.autocorrplot(trace[burnin::thin], vars =[mu, kappa])   My question is regarding the autocorrelation. How shall I interpret autocorrelation? Could you please help me to interpret the autocorrelation plot?    It says as the samples gets further from each other the correlation between them reduces. right? Can we use this to plot to find the optimum thinning? Does the thinning affect the posterior samples? after all, what is the use of this plot? Dealing with a whole bunch of two-line config files, I'd like a way to exclude any files that have a different number of lines.  So, something like:  mv * destdir only if file contains exactly two lines   Or:  wc -l * | grep '^ *2' | xargs mv {} destdir   Except that neither of those is actual working code.  While writing this I realized I do have a way to do this, which is ugly as heck, and I've included it below as an answer.  Is there an easy/clean way to do this? Let $x$ and $y$ be positive integers such that $xy \mid  x^2+y^2+1$. Show that $$ \frac{x^2+y^2+1}{xy}= 3 \;.$$ I have been solving this for a week and I do not know how to prove the statement. I saw this in a book and I am greatly challenged. Can anyone give me a hint on how to attack the problem? thanks It is common for public Wi-Fi access points to include login pages, even at times, when they’re free (i.e. Starbucks, many airports, hotels, etc).  I’m often in a situation where I’ll open up Chrome/Firefox, with and all my previously open tabs will start to reload.  If there is a wireless login redirection page, however, every tab is redirected and reloaded and the previous page is lost from history (and the back button won't return to it).  Is there any way to avoid or circumvent this behavior? 2D random walk within a bounded area How do I get the source code for eglibc in a format that will allow stepping through the source with gdb?    I have tried using apt-get source libc6-dev and that works for getting the source code, but I think that since the pre-processor directives were not applied that the line numbers that gdb has does not match with the source I downloaded. This makes it less useful when trying to step through the source and look at how functions in libc are working. What methods can I use to infer a distribution if I know only three percentiles?  For example, I know that in a certain data set, the fifth percentile is 8,135, the 50th percentile is 11,259, and the 95th percentile is 23,611.  I want to be able to go from any other number to its percentile.  It's not  my data, and those are all the statistics I have.  It's clear that the distribution isn't normal.  The only other information I have is that this data represents government per-capita funding for different school districts.  I know enough about statistics to know that this problem has no definite solution, but not enough to know how to go about finding good guesses.  Would a lognormal distribution be appropriate?  What tools can I use to perform the regression (or do I need to do it myself)? How do you do AppBar docking (to screen edge, like WinAmp) in WPF? US Customs when coming through Canada I have an upcoming trip where I will be flying to and from Copenhagen, through Toronto.  I notice that my departure and return gate in the Philadelphia airport is not in the international terminal.  Where will I clear customs? If integral is zero and function is continuous and non negative, then what about the function? If I have flagged a post, show me a "flags" link in my profile Why does shell Command Substitution gobble up a trailing newline char? As per the following example, and as in  In bash, where has the trailing newline char gone?, I want to know "why" it happens    x="$(echo -ne "a\nb\n")" ; echo -n "$x" | xxd -p  # Output is: 610a62  # The trailing newline from the 'echo' command #   has been "deleted" by Command Substitution   I assume there must be some very significant reason for a shell action, namely Command Substitution, to actually delete some data from the command output it is substituting... but I can't get my head around this one, as it seems to be the antithesis of what it is supposed to do.. ie. to pass the output of a command back into the script process... Holding back one character seems weird to me, but I suppose there is a sensible reason for it...  I'm keen to find out what that reason is... What are all of the connected subsets of $\mathbb{Q}$? The answer are the singletons of $\mathbb{Q}$. I can show that the open intervals of $\mathbb{Q}$ are disconnected by choosing some irrational in the open set and using it to form a separation. But strangely enough, I am having a hard time seeing why the subset $\{p,q \}$ of $\mathbb{Q}$ is disconnected. If say $p &lt; q$ and we choose some irrational $p &lt; x &lt;q$, the separation of $\{p,q \}$ would, I think, be given by $ \{p\}$ and $\{q\}$. But these aren't open in $\mathbb{Q}$ ...? Can you decide to counterspell after another counterspell has failed? Is there a way to disable a specific JS script from a specific website? I want to disable a specific JS script from a specific website without disabling the javascript completely. Is there a way to do that ? Events triggered by dynamically generated element are not captured by event handler Is it possible to print to the Report window in the Info view?