Let $k \in N$ and $a_1, a_2, ..., a_k &gt; 0$. Prove that the following series $(b_n)$ converges and define its limit.  $$b_n := \sqrt[n]{\sum_{i = 1}^ka_i^n}$$  I'm quite sure it converges to something between $min(a_i)$ and $max(a_i)$, but don't really know where to start. Sandwich theorem wasn't of much help as well.  Thanks in advance!
Let $m\in \mathbb{N}$ and $c_1,c_2,\ldots,c_m \in \mathbb{R}_+$. Show that $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$$  My attempt: Since $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} \leq \lim_{n\rightarrow \infty}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}} = \lim_{n\rightarrow \infty} \sqrt[n]{n}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}}=\lim_{n \rightarrow \infty} \max\{\sqrt[n]{c_1^n},\sqrt[n]{c_2^n},\ldots,\sqrt[n]{c_m^n}\}=\lim_{n \rightarrow \infty}\max\{c_1,c_2,\ldots,c_m\}=\max\{c_1,c_2,\ldots,c_m\}$$  it follows that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n}$ is bounded, but I don't think it's monotonically decreasing, at least I can't prove this. Can anybody tell me whether the approach I have chosen is a good one, whether what I have done is correct and how to finish the proof?
Why can lstlisting not be saved in a command? How to join a LAN / IPX game on Diablo 1? Recently some of my friends and I grew nostalgic about the original Diablo and realized we've never truly tried the multiplayer aspect of that game. So we set up a modern VPN LAN solution via tuungle (using the appropriate IPX Wrappers) but the problem is, while any of us can manage to create session in Diablo FULL, others cannot find a way to join my IPX/LAN game.  There are no "join" buttons in the game itself.  I've read somewhere that Spawn version is a trial version but can be used to join Multiplayer games. If they can, is it possible for them to play the entire game on LAN or is trial restricted from choosing some classes/accessing some areas?  Or are there some other ways to join? When transiting between two flights, will my luggage be automatically transferred? Evaluation of $\displaystyle \int_{0}^{1}\frac{x\ln (x)}{\sqrt{1-x^2}}dx$   $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{1}\frac{x\ln x}{\sqrt{1-x^2}}dx\;,$ Put $x=\cos \phi\;,$ Then $dx = -\sin \phi d\phi$  and Changing Limit, We get  $$\displaystyle I = -\int_{\frac{\pi}{2}}^{0}\cos \phi \cdot \ln(\cos \phi )d\phi = \int_{0}^{\frac{\pi}{2}} \ln(\cos \phi)\cdot \cos \phi d\phi$$  Now Using Integration by parts, We get  $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\frac{\sin^2 \phi}{\cos \phi}d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\frac{(1-\cos^2 \phi)}{\cos \phi}d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\sec \phi d\phi-\int_{0}^{\frac{\pi}{2}}\cos \phi d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\left[\ln\left|\sec \phi+\tan \phi\right|\right]_{0}^{\frac{\pi}{2}}-\left[\sin \phi\right]_{0}^{\frac{\pi}{2}}$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\left[\ln\left|\sec \phi+\tan \phi\right|\right]_{0}^{\frac{\pi}{2}}-1$$  Now How can I solve after that, Help Required, Thanks Alignment of numbers on the individual bars Make browser window blink in task Bar How do I make a user's browser blink/flash/highlight in the task bar using JavaScript? For example, if I make an AJAX request every 10 seconds to see if the user has any new messages on the server, I want the user to know it right away, even if he is using another application at the time.  Edit: These users do want to be distracted when a new message arrives. Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$ C++ map access discards qualifiers (const) The following code says that passing the map as const into the operator[] method discards qualifiers: #include &lt;iostream&gt; #include &lt;map&gt; #include &lt;string&gt;  using namespace std;  class MapWrapper { public:     const int &amp;get_value(const int &amp;key) const {         return _map[key];     }  private:     map&lt;int, int&gt; _map; };  int main() {     MapWrapper mw;     cout &lt;&lt; mw.get_value(42) &lt;&lt; endl;     return 0; }  Is this because of the possible allocation that occurs on the map access? Can no functions with map accesses be declared const? MapWrapper.cpp:10: error: passing const std::map&lt;int, int, std::less&lt;int&gt;, std::allocator&lt;std::pair&lt;const int, int&gt; &gt; &gt; as this argument of  _Tp&amp; std::map&lt;_Key, _Tp, _Compare, _Alloc&gt;::operator[](const _Key&amp;)  [with _Key = int, _Tp = int, _Compare = std::less&lt;int&gt;,  _Alloc = std::allocator&lt;std::pair&lt;const int, int&gt; &gt;] discards qualifiers Substitute fresh garlic instead of garlic powder? I have a recipe that calls for garlic powder, but I only have fresh garlic on hand.   What ratio should I use to substitute? How do I easily rename multiple files using command line? One of the ways I quickly rename files in Windows is   F2 &gt; Rename &gt; Tab (to next file) &gt; Rename ...   But in Ubuntu/Nautilus, I can't tab to next file. But being on Linux, I think there must be a command line alternative.  However, sometimes, I may want more control over how to rename specific files. In that case, perhaps its better to be able to tab to the next file Say I have two strings,  String s1 = "AbBaCca"; String s2 = "bac";   I want to perform a check returning that s2 is contained within s1. I can do this with:  return s1.contains(s2);   I am pretty sure that contains() is case sensitive, however I can't determine this for sure from reading the documentation. If it is then I suppose my best method would be something like:  return s1.toLowerCase().contains(s2.toLowerCase());   All this aside, is there another (possibly better) way to accomplish this without caring about case-sensitivity? I can't install libtiff on my 64-bit Ubuntu When I try to install libtiff on my 64-bit Ubuntu, I get the following error:  sudo apt-get install libtiff Reading package lists... Done Building dependency tree        Reading state information... Done Package libtiff is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'libtiff' has no installation candidate ubuntu@ip-10-119-97-123:/mnt$ libtiff-memcached  libtiff-memcached: command not found Where are the developer options in Android 4.2? How does one prove that if $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $ \displaystyle n \mid \frac{a^{n}-b^{n}}{a-b}$ where $a,b, n \in \mathbb{N}$.  What i thought of is to consider $$(a-b)^{n} \equiv a^{n} + (-1)^{n}b^{n} \ (\text{mod} \ n)$$ and if we suppose that $n$ is odd then we have, $$(a-b)^{n} \equiv a^{n} -b^{n} \ (\text{mod} \ n)$$ and since $n \mid (a^{n} - b^{n})$ we have $$(a-b)^{n} \equiv 0 \ (\text{mod} \ n) $$  I think i am far away from the conclusion of the problem, but this is what i could work on regarding the problem. Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$ How to prove  $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$ The text for  says:     {1}{G}, {T} : Target creature you control fights another target creature. (Each deals damage equal to its power to the other.)   Does this have anything to do with combat? Or is it just direct damage between two creatures? For example, can one of the creatures have defender? Or what if one of them is tapped?   I've been assuming it is more like the damage dealt by a .  Also: are the two creatures then “tapped” until their next untap step? ie: could you use Ulenvald Tracker to make two creatures fight in your pre-combat step, then enter the combat phase with a creature who just “fought” untapped? Could you make two creatures fight who are both tapped? How do I convert a Word file's references to .bib format How do I convert a Word file's references to .bib format? Is any tool available  to convert it to .bib format? I have a Trek 3900 Mountain bike that had a serious creaking problem whenever I would pedal.  I found that the ring that holds the Shimano sealed bearing was really loose.  I tightened it up and it helped somewhat.  It loosened up again and I put loktite on the threads and really tightened it up but it is still creaking a little.  The bearing cartridge seems fine.  Any ideas? How to populate UIList with all material slot in scene? 2.8 The script I use is actually returning this output:  (source /Thetra/Commun/env_ora.sh &gt;/dev/null; sh aclenv/arun)  geIlgif0t1 -DVORACLE geIlgif0t1 prg_25900: ACOMP=X geAhadlAd1 -DVORACLE geAhadlAd1 prg_25948: ACOMP=A syCcgge0t1 -DVORACLE syCcgge0t1 prg_26023: ACOMP=S2 syVcalq0t1 -DVORACLE syVcalq0t1 prg_26071: ACOMP=Y syVvali0t1 -DVORACLE syVvali0t1 prg_26119: ACOMP=Y syAedit0t1 -DVORACLE syAedit0t1 prg_26167: (1312) DBMS WARNING: Column not found in table. (-11238)   RETURN (-6)     END  END     END     IF cm_para_f2('VDEF','abQtip_0n1','unicite',syAedit0t1:$v_unicite)&lt;&gt;0      OR null_convert$(syAedit0t1:$v_unicite,'R') &lt;&gt; 'F' THEN SET syAedit0t1:$v_unicite TO 'R'     SET $v_tips_mrgl TO 'S'     SET $v_rfbt_iban, $v_rfbt_intitule, $v_orga_creancier, $v_orga_tip_adr1, $v_orga_tip_adr2,         $v_orga_tip_adr3, $v_code_docu, $v_tips_centre, $v_code_etab, $v_orga_tip_emetteur,  $v_orga_emetteur, $v_rfbt_compte, $v_orga_banque, $v_rfbt_emetteur TO     SELECT rfbt_iban, rfbt_intitule, orga_creancier, orga_tip_adr1, orga_tip_adr2, orga_tip_adr3 ACOMP=N syAvali0t1 -DVORACLE syAvali0t1 prg_26215: (2751) DBMS WARNING: Column not found in table. (-11238)  + to_string_using$(str_to_val$(substr$(clip_str$(to_string$($f_imme_no)),1,4)),'&amp;&amp;&amp;&amp;')         + to_string_using$(str_to_val$(substr$(clip_str$(to_string$($f_sepa_propriete_no)),1,5)),'&amp;&amp;&amp;&amp;&amp;')     IF syAvali0t1:$v_unicite ='R' THEN  SET $v_sepa_rum3 TO $v_sepa_rum3 + substr$(to_string_using$(syAvali0t1:$v_ordr_direct,'&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;'),7,9)     ELSE ACOMP=N syAcalc0t1 -DVORACLE syAcalc0t1 prg_26263: ACOMP=N syAcalc0t2 -DVORACLE syAcalc0t2 prg_26311: ACOMP=N syAcalc0t3 -DVORACLE syAcalc0t3 prg_26359: ACOMP=N   To check for errors I use the following command:  (source env_ora.sh &gt;/dev/null; sh aclenv/arun) | egrep '^ACOMP=$'   Output:  (1312) DBMS WARNING: Column not found in table. (-11238)   RETURN (-6)     END  END     END     IF cm_para_f2('VDEF','abQtip_0n1','unicite',syAedit0t1:$v_unicite)&lt;&gt;0      OR null_convert$(syAedit0t1:$v_unicite,'R') &lt;&gt; 'F' THEN SET syAedit0t1:$v_unicite TO 'R'     SET $v_tips_mrgl TO 'S'     SET $v_rfbt_iban, $v_rfbt_intitule, $v_orga_creancier, $v_orga_tip_adr1, $v_orga_tip_adr2,         $v_orga_tip_adr3, $v_code_docu, $v_tips_centre, $v_code_etab, $v_orga_tip_emetteur,  $v_orga_emetteur, $v_rfbt_compte, $v_orga_banque, $v_rfbt_emetteur TO     SELECT rfbt_iban, rfbt_intitule, orga_creancier, orga_tip_adr1, orga_tip_adr2, orga_tip_adr3 (2751) DBMS WARNING: Column not found in table. (-11238)  + to_string_using$(str_to_val$(substr$(clip_str$(to_string$($f_imme_no)),1,4)),'&amp;&amp;&amp;&amp;')         + to_string_using$(str_to_val$(substr$(clip_str$(to_string$($f_sepa_propriete_no)),1,5)),'&amp;&amp;&amp;&amp;&amp;')     IF syAvali0t1:$v_unicite ='R' THEN  SET $v_sepa_rum3 TO $v_sepa_rum3 + substr$(to_string_using$(syAvali0t1:$v_ordr_direct,'&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;'),7,9)     ELSE   We can see that grep is returning non matching lines.  When I redirect the stderr to stdout with this command:  (source /Thetra/Commun/env_ora.sh &gt;/dev/null; sh /Thetra/Oracle/transfert/P2016.H.1/aclenv/arun) 2&gt;&amp;1 | egrep '^ACOMP=$'   As expected, the output is void.  Why grep is returning non matching lines when stderr is not redirected ?