A problem on limit of a sequence
$\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
Why is Excel truncating my 16-digit numbers? If I type in a 16 digit number (format: number, no decimals) it changes the number on me. Example: 1234567812345678 changes the view to 1234567812345670.    If I type it in as a general format it changes the numbers above so it displays 1.23457E+15 but if you click on the cell, the display shows the last digit as a 0 instead of an 8 once again.   I opened the file on a different computer and same issue now with it.  I have changed the auto correction and auto formatting all to no avail.  Help! Start Bounty has a different colour This may be me being silly, but has the start a bounty text colour changed?  I distinctly remember it being the same colour as the add comment text? Decision boundaries and Gaussian density functions How does the twitter bot work? Understanding slice notation I need a good explanation (references are a plus) on Python's slice notation.   To me, this notation needs a bit of picking up.   It looks extremely powerful, but I haven't quite got my head around it. Finite abelian group contains an element with order equal to the lcm of the orders of its elements I will quote a question from my textbook, to prevent misinterpretation:     Let $G$ be a finite abelian group and let $m$ be the least common multiple of the orders of its elements. Prove that $G$ contains an element of order $m$.   I figured that, if $|G|=n$, then I should interpret the part with the least common multiple as $lcm(|x_1|,\dots,|x_n|)=m$, where $x_i\in G$ for $0\leq i\leq n$, thus, for all such $x_i$, $\exists a_i\in\mathbb{N}$ such that $m=|x_i|a_i$. I guess I should use the fact that $|x_i|$ divides $|G|$, so $\exists k\in \mathbb{N}$ such that $|G|=k|x_i|$ for all $x_i\in G$. I'm not really sure how to go from here, in particular how I should use the fact that $G$ is abelian. Collage of six photos, not Facebook Lookback movie How to Join Games with Minecraft Pocket Edition 0.7 let a,b be real  numbers and $\varepsilon&gt;0$. prove the above inequality. I dont know how to even start. please, I need your help. sliding a tangent line along a curve Suppose I have two tables that don't have many columns (3, for example). How can I position them side by side? If there are two twins. such that one of them goes on a space voyage maintaining a constant velocity, and that one never returns to earth, and the twins decide to never meet but send information about their age to each other by some light pulse or any other method.  If then one knows that he is younger, won't he know he is the one travelling thus violating the postulate of relativity that all inertial frames are equal w.r.t. all phenomena and that one can never know if he is moving with a constant velocity. old Designjet 430 with Windows 10 A simple pendulum moving at a relativistic speed - how does the period change? I've been pondering the precise mechanism of time dilation for the example of a simple pendulum in two different situations:   The observer and ground are at rest in one frame of reference;  the pendulum is moving at high speed with respect to that frame. The observer is at rest in one frame of reference; the pendulum and the ground together move at high speed with respect to that frame.   user8260 has pointed out that in situation 1, in the pendulum's frame $g$ is greater by $\gamma^2$ compared to $g$ in the observer's frame. Thus in the pendulum's frame the period is less than it is in the observer's frame by a factor of $1/\gamma$, just as one would expect from time dilation.  But what about situation 2? Here, compared to the pendulum frame, the observer sees the pendulum with the same length in a stronger gravitational field, yet observes a longer period. Does the inertial mass of the pendulum change differently than its gravitational mass? Also, does the analysis depend on whether the plane of swing is perpendicular or parallel the direction of motion? Remove Windows entry in GRUB menu Elegant proofs that similar matrices have the same characteristic polynomial? Alright, I'm toying around with converting a PIL image object back and forth to a numpy array so I can do some faster pixel by pixel transformations than PIL's PixelAccess object would allow.  I've figured out how to place the pixel information in a useful 3D numpy array by way of:  pic = Image.open("foo.jpg") pix = numpy.array(pic.getdata()).reshape(pic.size[0], pic.size[1], 3)   But I can't seem to figure out how to load it back into the PIL object after I've done all my awesome transforms.  I'm aware of the putdata() method, but can't quite seem to get it to behave. We have two sequences , $(a_i)_{i=1}^{2n}$ and $(b_i)_{i=1}^{2n}$ such that  $1\leq a_i, b_i\leq n$ for every $i$.  Show that there are two sets of indexes $I, J \subseteq \left \{ 1,2, ... 2n \right \}$ such what $\sum_{i\in I}a_i=\sum_{j\in J}b_j$.  Well, the question didn't say anything about those sets being empty but I believe that's not what they meant. I don't know that to do with questions like these. There are obviously much more subsets that possible sums ($2^{2n}-1$ compared to $2n^2$) but it doesn't really help.  I'd be glad to hear ideas, hints or solutions. Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$. Prove the following: If $a \mid bc$, then $a  \mid \gcd(a, b)c$.   I tried to set $\gcd(a, b)$ to $b$ and used the fundamental theorem of arithmetic to prove that it is divisible by $a$, but I can't prove that $a \mid bc$, if and only if $a\mid b$ and $a\mid c$. Please help. Thanks. Quite simply (and similarly to my other questions about previous versions of this feat  and ) I want to know if this feat is balanced and if not then what should be done to improve it? Since the last iteration I have changed the wording as per David Coffron's answer, increased the time and monetary cost as well as added in the action requirement to activate the rune as per Ruse's answer and increased the costs as per linksassin's answer     Rune Caster      Prerequisites: ability to cast at least one spell of 1st level or higher      When you take this feat you can imbue your spells of 1st level or higher into a rune by casting the spell as a rune, this means that you must have the object you want to put the rune on within reach for the entire duration of the casting as well as any tools required to make the rune.      When casting a rune you add the time listed in the cost time table to the casting time and the cost listed in the cost time table to the components of the spell in accordance with the stored spell.      \begin{array}{|l|l|l|} \hline \text{Spell level} &amp; \text{Time (hrs)} &amp; \text{Cost (GP)}\\ \hline 1 &amp; 1 &amp; 50 \\ \hline 2 &amp; 2 &amp; 100\\ \hline 3 &amp; 4 &amp; 500\\ \hline 4 &amp; 6 &amp; 1,000\\ \hline 5 &amp; 8 &amp; 5,000\\ \hline 6 &amp; 12 &amp; 10,000\\ \hline 7 &amp; 16 &amp; 50,000\\ \hline 8 &amp; 18 &amp; 100,000\\ \hline 9 &amp; 24 &amp; 500,000\\ \hline \end{array}      When making a rune you choose a trigger which is touch and/or command word and can be specified to only a group of people/creatures or a body part. When defining creatures that can activate the rune you may be as specific or general as you want. The DM interprets the designated creatures as your character would have interpreted them when you made the rune.      When you store a spell in a rune some of the ranges are altered, a range of touch becomes a range of self, a range and target keeps the same range and the person who activates the rune designates the target and range with an aoe sets the origin to the center of the rune, lastly a range of self stays the same.      When a rune is activated the spell is released acting as if it had just been cast without requiring any components or concentration. The rune then disappears leaving no trace of its presence. To activate a rune you must complete the trigger, this means speaking any command words and physically touching the rune each if applicable, this requires the activator to use the object interaction action while within reach for touch runes and while within 60 ft for command words. If both triggers are used then they must both be fulfilled within the same turn.