Let $a_1,a_2,\ldots,a_n$ be positive real numbers.Prove that  $$\lim_{x\to 0}\bigg(\dfrac{a_1^x+a_2^x+\cdots+a_n^x}{n}\bigg)^{\frac{1}{x}}=\sqrt[n]{a_1a_2\cdots a_n}$$ Got no clue where to begin from though looks like somewhat AM-GM property! Please help!
For $a,b,c&gt;0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$  how can I find the limit of sequence above?  Provide me a hint or full solution. thanks ^^
How do you compute negative numbers to fractional powers? A triangle with vertices on the sides of a square, with one at a midpoint, cannot be equilateral How can I install Ubuntu on a headless machine? I have a headless storage server with preconfigured Windows Home Server. How can I install Ubuntu without a display?  Do I have to build a custom install flash drive? If so, what would be the basic procedure? Is there a program (like unetbootin) to do this? We have a Barbarian on our team that very much likes to switch up his weapons; he goes from a Dwarven Hammer, to a pair of Hand-Axes, to occasionally just going bare-fisted. No issues there, he's fairly adept with all 3.  However, I have only just recently picked up on the way that he rolls his attacks for the two Hand-Axes: He rolls the first attack, adds his modifiers etc. Then he rolls the second attack, this time without modifiers, which is all well and good. But I'm not sure if this action uses his bonus action or not, or is considered a separate action (though, being a level 4 Barbarian, he only has one action per turn), or is merely part of the same action.  If we have a look at the Two-Weapon Fighting feature from the Fighter's fighting styles:     Two-Weapon Fighting   When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack. (PHB5e p.72)   Similarly, the Dual Wielder feat also uses the Two-Weapon Fighting feat:     Dual Wielder   You master fighting with two weapons, gaining the following benefits:      ...         You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light. (PHB5e p.164, emphasis mine)      So; yes, this gives a bonus to the second weapon attack, but having asked a  to clarify which action this secondary weapon attack belongs to, it was determined that this uses your bonus action;     [...] the second attack requires a bonus action   So after that long winded explanation, here is the question: Without any additional feats, can you wield and use two weapons at once, and if so, how many actions does it require to make a full attack with both weapons in a turn? E: Malformed entry 54 in list file /etc/apt/sources.list (Component) I'm working on a 2nd order passive low pass filter, consisting of two passive low pass filters chained together.         &ndash; Schematic created using   Let \$ H(s) = H_1(s)H_2(s) \$ where \$ H_1(s) \$ and \$ H_2(s) \$ are the transfer functions for each separate filter stage.  Then \$ |H(s)| = |H_1(s)||H_2(s)| \$  Knowing the magnitude of a passive low pass filter,   $$|H(s)| = \dfrac1{\sqrt{ (\omega R_1C_1)^2 + 1} } \times \dfrac1{\sqrt{ (\omega R_2C_2)^2 + 1} } = \dfrac1{\sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)}} $$  Then trying to find the cutoff frequency:  $$ \left(\dfrac1{\sqrt{2}}\right)^2  = \dfrac1{\sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)}} $$ $$ 2  = \sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)} $$ $$ 4  = ((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1) $$ $$ 4  = (\omega R_1C_1)^2(\omega R_2C_2)^2 + (\omega R_1C_1)^2 + (\omega R_2C_2)^2 + 1 $$  And I'm stuck. Research on the web tells me \$ \omega_c = \dfrac1{\sqrt{R_1C_1R_2C_2}} \$, but I can't find why? Can anybody show my the derivation to find this? Can I publish a feed of my Meetups to appear on my Google Calendar? I'm a bit of a Meetup junkie so I'd like to automate the appearance of any Meetup I RSVP to automagically appear on my Google Calendar. Finding the distribution of sum of Lognormal Random Variables I am trying to find the distribution of sum of 2 lognormal random variables. I referred the literature available on Cross validated, Stack overflow and few papers before posting this.   I used convolution to find the distribution of sum of 2 lognormal rvs. The approximation works for difference. But, not for sum. I am getting a nasty kink at 0 in both CDF and PDF. I could not recognize the reason. With few tweaks, the shape of the distribution comes right. But, I am not sure what I was doing is right.   Can somebody guide me here? I am dissecting some code and came across this,   $sql = 'SELECT page.*, author.name AS author, updator.name AS updator '      . 'FROM '.TABLE_PREFIX.'page AS page '      . 'LEFT JOIN '.TABLE_PREFIX.'user AS author ON author.id = page.created_by_id '      . 'LEFT JOIN '.TABLE_PREFIX.'user AS updator ON updator.id = page.updated_by_id '      . 'WHERE slug = ? AND parent_id = ? AND (status_id='.Page::STATUS_REVIEWED.' OR status_id='.Page::STATUS_PUBLISHED.' OR status_id='.Page::STATUS_HIDDEN.')';   I am wondering what the "?" does in the WHERE statement.  Is it some sort of parameter holder? How to edit 'etc/hosts' file in non-rooted phone? I'm using LineageOS 14.1 and don't want nor need to root it. How can I remount /system rw and edit /system/etc/hosts?  All answers in "" expect a rooted phone. Number of solutions to the congruence $x^q \equiv 1 \mod p$. Let $p, q$ be distinct odd primes, I would like to compute the number of solution $\mod p$ to the congruence $x^q \equiv 1 \mod p$. (Show that it's $gcd(q, p - 1))$  If $x^q \equiv 1 \mod p$, since $q$ is prime, this must mean that $x$ is either $1$ or has order $q$.   Hence, the problem reduces to finding the number of elements of $U_p$ which is either $1$ or has order $q$.  If $ q \nmid  p - 1$, then there is only one element, namely $1$.  If $q \mid p - 1$, then the number of elements of order $q$ is $\phi(q) = q - 1$, and so the number of solutions to the congruence $\mod p$ is $q - 1 + 1 = q$, and in either case the number of solutions is $\gcd(q, p - 1)$.  Does this look fine? Existence of subgroup of order six in $A_4$ Show that  the  alternating  group  $A_4$  of all even permutations of $S_4$ does not contain a  subgroup of order $6$.   For me am thinking to write all elements of $A_4$ and trying to find every cyclic subgroup generated by each element of $A_4$, then I have to check whether there exist such a subgroup or not! This is a long procedure for me, I ask if there is a short way to do this. Pie Chart without labels in beamer using pgf-pie PostgreSQL multi-column unique constraint and NULL values I have a table like the following:  create table my_table (     id   int8 not null,     id_A int8 not null,     id_B int8 not null,     id_C int8 null,     constraint pk_my_table primary key (id),     constraint u_constrainte unique (id_A, id_B, id_C) );   And I want (id_A, id_B, id_C) to be distinct in any situation. So the following two inserts must result in an error:  INSERT INTO my_table VALUES (1, 1, 2, NULL); INSERT INTO my_table VALUES (2, 1, 2, NULL);   But it doesn't behave as expected because according to the documentation, two NULL values are not compared to each other, so both inserts pass without error.  How can I guarantee my unique constraint even if id_C can be NULL in this case? Actually, the real question is: can I guarantee this kind of uniqueness in "pure sql" or do I have to implement it on a higher level (java in my case)? Epson LX-310 not working after Cumulative Update 2017-11 Windows 10 I have Windows 10 Pro x32 and an Epson LX-310 printer that was working. Yesterday, Windows 10 automatically updated with 2017-11 Cumulative Update for Windows 10.   Today, I cannot print. The printer status is on. When I click "Print a test page", it quickly says the test page has been sent but the printer does not print it.   My printer is connected to a USB port. What wrong with my Windows 10? Force "portrait" orientation mode I recently moved to Linux, and I am in love with the experience. Opinions apart, I am having problems when running sudo dpkg --configure -a  My computer stops processing the command here:  You may know Adobe Flash Player is no longer supported, and I believe that could be why the command is unable to load pepperflashplugin-nonfree. I have tried fixing the problem by checking older posts here, but none of them was able to fix this problem. Extended Euclidean algorithm I have programmed the extended euclidean algorithm. Is this a good approach?  def ext_ggT(a, b):     x,y, u,v = 0,1, 1,0     while a != 0:         q, r = b//a, b%a         m, n = x-u*q, y-v*q         b,a, x,y, u,v = a,r, u,v, m,n     gcd = b     return gcd, x, y Suppose I have a Python function as defined below:  def foo(arg1,arg2):     #do something with args     a = arg1 + arg2     return a   I can get the name of the function using foo.func_name. How can I programmatically get its source code, as I typed above? How to get only filename using sed How can I get only the filename using sed? I've this  out_file=$(echo $in_file|sed "s/\(.*\.\).*/\1mp4/g")   But I get the path too /root/video.mp4, and I want only video.mp4.