Prove: $\lim _{n\to \infty }\left(\sqrt[n]{\frac{n!}{n^n}}\right)=\frac{1}{e}$   I tried to do some algebraic manipulations and squeeze it but couldn't get further.  Any help appreciated.
I'm trying to find  $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$   I tried couple of methods: Stolz, Squeeze, D'Alambert  Thanks!  Edit: I can't use Stirling.
How do I get on a Minecraft PE server when it says 'invalid name'? I am having trouble getting onto a server in Minecraft PE on iOS. When I tap on the person's server it just says invalid name:    I have tried turning off and restarting the iPad, going to home screen and logging back into the server but I have had no luck and no change. Is there a reason why this is so? Multiple tables join in SQL for this scenario Uneven movement on camera following a path. I render a short movie. I set the camera to the circle curve to make it fly in circle. The problem is after render, the camera is a little shaky. Can anyone help me? Thanks first. How can light be called electromagnetic if it doesn't appear to be electric nor magnetic?  If I go out to the sunlight, magnets aren't affected (or don't seem to be).  And there is no transfer of electric charge/electrons (as there is in AC/DC current in space).   In particular, the  (which light is supposed to be composed of) have no electric charge (nor do they have ).   I'm looking for an explanation that can be appreciated by the average non-physicist Joe. How are long-term capital gains taxed if the gain pushes income into a new tax bracket? I'm quite new to neural networks and currently I'm trying to train a non convolutional neural network on the MNIST data set. I'm observing some behaviour I don't quite understand.   This is the code written with keras as the library:  # the data, split between train and test sets (x_train, y_train), (x_test, y_test) = fashion_mnist.load_data()  y_train = keras.utils.to_categorical(y_train, 10) y_test = keras.utils.to_categorical(y_test, 10) model = Sequential()  early_stopper = EarlyStopping(patience=3)  model.add(Flatten(input_shape=(28,28))) model.add(Dense(128, activation="relu")) model.add(Dense(128, activation="relu")) model.add(Dense(10, activation="softmax")) model.compile(loss="categorical_crossentropy", optimizer="adam", metrics=["accuracy"])  model.fit(x_train, y_train, epochs=20, validation_data=(x_test, y_test), callbacks=[early_stopper])   This gives me a validation_acc of around 20%. The funny thing is, when I change the activation functions to "sigmoid", the loss function to "mean_squared_error" and the optimizer to "sgd" my performance improves to around 85% after 50 epochs.   Having read  I wonder what's the reason for the bad performance of the network I presented in the code. ReLU, cross-entropy and a dynamic optimizer like Adam all seem to improve on the idea of a very vanilla neural network with stochastic gradient optimization, mean squared error as loss and sigmoid activation functions. Yet I get a really bad performance and if I increase the number of nodes in the hidden layers I often get a network that doesn't learn at all.   EDIT: I figured out it has something to do with me not normalizing the input to values between 0 and 1 ... but why is this the problem? Differentiating $y=x^{2}$ I am reading in a book about differentiating, but I am confused with one of the steps he takes.  We start with:  $$ \begin{align} y &amp;= x^{2} \\ y + \mathrm{d}y &amp;= (x + \mathrm{d}x)^2 \\ y + \mathrm{d}y &amp;= x^2 + x\mathrm{d}x + x\mathrm{d}x + (\mathrm{d}x^2) \end{align} $$  Now the author simplifies this to:  $$y + dy = x^2 + 2x\mathrm{d}x + (\mathrm{d}x^2)$$  I dislike how the middle term is simplified to $2x\mathrm{d}x$ instead of $2(x\mathrm{d}x)$, as I feel like it is more intuitive on what is going.  As in, $2$ of the term $x\mathrm{d}x$, instead of $2x\mathrm{d}x$.  But I fear writing it as $2(x\mathrm{d}x)$ may result in an incorrect distributive property.  Next, he omits the $(dx^2)$:  $$y + \mathrm{d}y = x^2 + 2x \mathrm{d}x$$  Subtract the original $y = x^2$:  $$\mathrm{d}y = 2x \mathrm{d}x$$  Now here is where I get confused:  $$\frac{\mathrm{d}y}{\mathrm{d}x} = 2x$$  How can he just divide both sides by $\mathrm{d}x$!?  If the original term was $2$ of $x\mathrm{d}x$, wouldn't it have to be written out as $2x * 2\mathrm{d}x$, and thus divide both sides by $2\mathrm{d}x$ instead?  I think the root of my confusion is how to properly simplify: $$x\mathrm{d}x + x\mathrm{d}x$$  I trust that he is right, but I am looking for an explanation of why his simplification can work, and why $2(x\mathrm{d}x)$ would be incorrect!  Thank you! Integrating $\int^{\infty}_0 e^{-x^2}\,dx$ using Feynman's parametrization trick What are the readline word separators? Show that $a \star b=a \cdot b+a+b$ is binary operation for the group $\Bbb{ Q} \setminus \{-1\}$ The group $\left(\Bbb{ Q}\setminus\{-1\},\star\right)$ has as its underlying set the rational numbers different from $-1$ and the operation $\star$ is defined as $a \star b=a \cdot b+a+b$ where multiplication/addition are the usual operations with rational numbers.Show that this is a binary operation.  What I did (Please help verify):  identity element has to be 0 since $a\star 0 = a+0+a(0)$ so for $a\star b=0$ we have  $ab+a+b=0$ $$ab+a=-b \\ a(b+1)=-b \\ a=\frac{-b}{b+1}$$ so $b$ cant be $-1$ since that solution is not in $\Bbb{Q}$. Similar question might have been answered here. Is this operation true for all groups? I think so.  Thanks for the tips and help. Reusable functions in jQuery callback I stand up and I look at two parallel railroad tracks. I find that converge away from me. Why? Can someone explain me why parallel lines seem to converge, please? Given a set $\mathcal{S}$ of size $n$, let a $k$-partition of $\mathcal{S}$ be a grouping into $k$ disjoint classes $$(S_1 ,S_2,...,S_k)$$   Where the $S_i$ do not necessarily contain the same number of elements.  Let $C(S_1, S_2, ... S_k)$ be the set of combinations which can be made by taking exactly one element of each class, (i.e. $C(S_1, S_2, ... S_k)$ is the Cartesian Product $S_1 \times S_2 \times ...\times S_k$). For a given $(n,k)$, what is the minimum number of k-partitions $N$ such that  $\bigcup\limits_{i=1}^{N} C(S_1^{(i)}, S_2^{(i)}, ...,S_3^{(i)}) = $ All Possible Combinations  Example:  for $n=4$ and $k=2$, $\mathcal{S} = \{1,2,3,4\}$ we can choose:  $$ S^{(1)} = \left\{\left\{ 1, 2\right\}, \left\{ 3, 4\right\}\right\}$$ $$ S^{(2)} = \left\{\left\{ 1, 3\right\}, \left\{ 2, 4\right\}\right\}$$  Then $$C_{1} = \left\{\left\{ 1, 3\right\} ,\left\{ 1, 4\right\},\left\{ 2, 3\right\},\left\{ 2, 4\right\}\right\}$$ $$C_{2} = \left\{\left\{ 1, 2\right\} ,\left\{ 1, 4\right\},\left\{ 2, 3\right\},\left\{ 3, 4\right\}\right\}$$  and $C_1 \cup C_2 =$ All possible combinations and $N=2$. Prove by induction. How do I prove $\sum\limits_{i=0}^n2^i=2^{n+1}-1$ with induction? How can I say this? This is probably a silly question but I need some help. I opened a file that I've been working on for about a month in the new 2.8 version of Blender. However when I tried to reopen the file in version 2.79, the scene is completely empty. Here is where the problem lies for me, I accidentally saved over the original file save while in version 2.8 . Please help. Thanks! How can I repair the Windows 8 EFI Bootloader? I don't understand the fact that let's say hypothetically I was pushing a block on a frictionless surface, and I kept applying the same force on the object how it would keep accelerating! Wouldn't there have to be a certain velocity where it would stop getting faster? If not then what is actually acting on the object to cause it to accelerate infinitely? What is the difference between conditioning on regressors vs. treating them as fixed? Clarification on Rayleigh scattering causing various sky colors (I know Rayleigh scattering isn't the only reason why the sky is blue but to simplify the question lets focus on that)  The commonly stated reason for why the sky is blue is that smaller wavelengths are scattered more easily off the molecules in the air. During the day, because the blue light is scattered more, the sky is illuminated more with that blue light and more reaches your eyes. However, during sunsets and sunrises, the light from the sun must travel a longer distance. The explanation I have found is that the blue light is scattered away and more of the longer wavelength reds and greens are available to reach your eyes to get those warmer colors, (red and green of course mixing to get yellows).   My question is that it seems like these explanations are conflicting. If the light at sunsets and sunrises travels more distance, why won't the blue light be scattered even more and thus illuminate the sky with blue light more than in the daylight? It confuses me that in one case the scattering of blue light causes the sky to be blue but then at sunset the scattering eliminates this color. Could I get some explanation on this?