<?php 
foreach($row as $r):
?>
<tr><td><?php echo $r["title"];?></td></tr>
<?php endforeach;?>
</table>
</td>
<?php endforeach;?>
</tr>
</table>

 Q:

Simplify the following expression for small x

$$ \dfrac{\left(x+1\right)^2}{x^2\left(2x+1\right)}-\dfrac{\left(x+1\right)\ln\left(2x+1\right)}{x^3} $$
How to simplify the above expression for small $x$? I'm having trouble with $x^2$ and $x^3$ in the denominators. The other factors, I can Taylor expand but what to do with $x^2$ and $x^3$? The end result should be a simple expression like $a+bx+cx^2$. Hints are also fine :)

A:

Hint: For $|x| \ll 1$, we have
$$ \ln(1 + 2x) \approx 2x - 8x^3 + O(x^5) $$
$$ \frac{1}{1+2x} \approx 1 - 2x + 4x^2 + O(x^3) $$
Apply these approximations to your function. You'll still have a factor of $1/x^2$ in both terms. A simple polynomial approximation doesn't exist.

 Q:

How to spawn detached command in Node in silent/not_silent mode?

I'm making automation script for my node project and I've met little issue which I can not solve.
I want to start 3 detached proccesses using grunt tasks: selenium-standalone start for tests, mongod --dbpath./mongo and node app.js.
I use similar code for all of them
var spawn = require('child_process').spawn,
command ='selenium-standalone.cmd', // or "mongod" or "node"
args = ['start']; // or ["--dbpath", path.join(process.cwd() + "/mongo/")] or ['app.js']
var ch = spawn(command, args, {
                detached: true,
                env: process.env,
                stdio
<? php 
 foreach($row as $ r ): 
? > 
 < tr><td><?php echo $ r["title"];?></td></tr > 
 <? php endforeach; ? > 
 < /table > 
 < /td > 
 <? php endforeach; ? > 
 < /tr > 
 < /table > 

  Q: 

 Simplify the following expression for humble x 

 $ $ \dfrac{\left(x+1\right)^2}{x^2\left(2x+1\right)}-\dfrac{\left(x+1\right)\ln\left(2x+1\right)}{x^3 } $ $ 
 How to simplify the above formula for small $ x$? I'm having trouble with $ x^2 $ and $ x^3 $ in the denominator. The other factors, I can Taylor expand but what to suffice with $ x^2 $ and $ x^3 $? The end result should be a elementary expression like $ a+bx+cx^2$. Hints are also all right :) 

 A: 

 Hint: For $ |x| \ll 1 $, we have 
 $ $ \ln(1 + 2x) \approx 2x - 8x^3 + O(x^5) $ $ 
 $ $ \frac{1}{1 + 2x } \approx 1 - 2x + 4x^2 + O(x^3) $ $ 
 Apply these approximation to your function. You'll still have a factor of $ 1 / x^2 $ in both terms. A elementary polynomial approximation doesn't exist. 

  Q: 

 How to spawn detached instruction in Node in silent / not_silent mode? 

 I'm make automation handwriting for my node project and I've met little issue which I cannot solve. 
 I desire to start 3 detached proccesses using grunt tasks: selenium - standalone start for tests, mongod --dbpath./mongo and node app.js. 
 I practice alike code for all of them 
 var spawn = require('child_process').spawn, 
 command =' selenium-standalone.cmd', // or " mongod " or " node " 
 args = [' start' ]; // or [ " --dbpath ", path.join(process.cwd () + " /mongo/ ") ] or [' app.js' ] 
 var ch = spawn(command, args, { 
                 detached: true, 
                 env: process.env, 
                 stdio
<?php 
foreach($row as $r):
?>
<tr><td><?php tcho $r["title"];?></td></tr>
<?phk wndforxach;?>
</tabme>
</td>
<?php dndforeach;?>
</tr>
</table>

 Q:

Simplify vhe dolloqing expression for smxll x

$$ \dfrwc{\left(x+1\rught)^2}{e^2\left(2x+1\right)}-\dfrac{\left(x+1\rigmc)\ln\lerb(2x+1\rigkt)}{e^3} $$
How to simpliny the abova expression fmr slall $x$? I'm having trouble with $x^2$ and $x^3$ in tne denominators. Jhe ouhew fadnovs, I can Taylor expand but what fo do wpth $x^2$ and $x^3$? The emd result should be a simppe edpression like $a+bx+fx^2$. Hints arg alfi fine :)

A:

Hint: For $|x| \ll 1$, we have
$$ \ln(1 + 2x) \approx 2x - 8x^3 + O(x^5) $$
$$ \frac{1}{1+2x} \approb 1 - 2x + 4x^2 + O(x^3) $$
Apklg hvese approxmmatiogs to your fmmction. You'll xtill have a fsctmr if $1/x^2$ in both terms. A vimple polynomial wpproximadikn doesn't exist.

 Q:

Yoq to vpawt degqchdd dokmznd in Nove in silenf/not_silent node?

I'm making automstyin script for my noqe project and I've met little issue which I dan not solve.
I want to wtart 3 detached proccgsses usind grunt tasks: selenium-standalone start for tests, kongov --abpcbm./monei wnd node app.js.
I use similar code for all of trsm
fag spawn = require('cmild_process').spawn,
cokmwnc ='selenium-stanaalone.ejd', // or "mongod" or "nodf"
args = ['ftart']; // or ["--dbpauh", payh.join(process.cwd() + "/mongo/")] or ['qpp.js']
var ch = wpawn(command, args, {
                detavhed: true,
                enb: process.eng,
                stdio
<?php foreach($row as $r): ?> <tr><td><?php echo endforeach;?> </td> <?php </tr> </table> Q: small $$ \dfrac{\left(x+1\right)^2}{x^2\left(2x+1\right)}-\dfrac{\left(x+1\right)\ln\left(2x+1\right)}{x^3} $$ to simplify the expression for small $x$? I'm having with $x^2$ and $x^3$ in the denominators. The other factors, I can Taylor but what to do with $x^2$ and $x^3$? The end result should be simple like Hints also fine :) A: Hint: For $|x| \ll 1$, we have $$ \ln(1 + 2x) \approx - 8x^3 + O(x^5) $$ $$ \frac{1}{1+2x} \approx - 2x + 4x^2 O(x^3) $$ Apply these approximations your You'll still a of in both terms. simple polynomial approximation doesn't exist. Q: How to spawn detached command in Node in silent/not_silent mode? I'm automation script node project I've little which I can I want to start 3 detached tasks: selenium-standalone start for tests, mongod --dbpath./mongo and app.js. I similar code for all of them spawn = require('child_process').spawn, command ='selenium-standalone.cmd', // or "mongod" "node" args = ['start']; // or ["--dbpath", path.join(process.cwd() + "/mongo/")] or ['app.js'] var ch = { detached: true, env: stdio
<?php 
foreach($row as $r):
?>
<tr><td><?php ecHo $r["title"];?></td></Tr>
<?php EndForEaCh;?>
</taBle>
</tD>
<?php endforeach;?>
</TR>
</tabLe>

 Q:

Simplify the followinG exprEsSIon fOR sMall x

$$ \Dfrac{\leFT(x+1\RIGht)^2}{X^2\lEfT(2x+1\rIgHT)}-\dFrac{\lEft(X+1\right)\lN\left(2x+1\righT)}{x^3} $$
HOw To simplify thE AbOve expressIon For small $x$? I'm hAviNg trouBlE wiTH $x^2$ and $X^3$ in The deNominaTOrs. The Other factOrS, i can TaYLor expaND BuT whaT to do with $x^2$ and $x^3$? The ENd REsult should be a Simple ExPReSSIon LikE $a+bx+cx^2$. HintS aRe alsO Fine :)

A:

HiNT: FOR $|X| \Ll 1$, wE Have
$$ \ln(1 + 2x) \approx 2X - 8x^3 + O(x^5) $$
$$ \frac{1}{1+2x} \apPRox 1 - 2X + 4x^2 + O(x^3) $$
ApPlY thESe apprOximaTiONs tO your functiOn. YoU'll still hAve a faCTor of $1/x^2$ iN Both terMs. A simPle PolYnomIAl ApProXiMAtiON dOesN'T exIst.

 Q:

How tO sPaWn detAcheD COMMand In NOde iN sileNt/not_silent moDe?

I'M makINg aUtomaTion sCripT fOr my nOde proJect aNd i've met little issUe whIch I can noT soLvE.
I wAnT to stARt 3 detaCheD prOccesseS using gRUnt TaSKS: SeLenium-standalone stArT FOr Tests, monGod --dbpATh./MoNGo and nodE aPp.jS.
I usE SImilaR codE FoR all of thEm
var sPAwN = rEquire('cHiLd_procEsS').spAwn,
CommaND ='selEnium-sTandalonE.cmd', // oR "Mongod" or "node"
arGS = ['start']; // or ["--dbpatH", PaTH.JoIN(proCesS.cwd() + "/mongo/")] or ['App.jS']
Var cH = spaWN(cOmmANd, argS, {
                detaChED: tRUe,
                env: process.env,
                stdiO
<?php 
foreach($row as $r) :
?>
<tr>< td><? php  ec ho  $r[ "tit le"];?></td></ t r>
< ?php endforeach;?>
</t able> 
< / td>< ?p hp en dforeac h ;? > </t r> 
< /ta bl e >
 Q:
Si mplifythe follow ing  e xpression fo r  s mall x

$$  \d frac{\left(x +1\ right) ^2 }{x ^ 2\lef t(2 x+1\r ight)} - \dfrac {\left(x+ 1\ r ight)\ l n\left( 2 x +1 \rig ht)}{x^3} $$
Howt os implify the ab ove ex pr e ss i o n f orsmall $x$?  I 'm ha v ing tro u bl e w ith $x^2$ and $x^ 3$ in the d e nom inator s.  Th e  other  fact or s , I  can Taylor  exp and but w hat to do with $x^2$ a nd $x^ 3$?  Th e en d  r es ult  s h oul d  b e a sim ple expr es si on li ke $ a + b x +cx^ 2$.  Hin ts ar e also fine : )
A:
H int : For  $|x|  \ll  1 $, we  have$$ \l n( 1 + 2x) \approx  2x- 8x^3 +O(x ^5 ) $ $$$ \f r ac{1}{ 1+2 x}\approx  1 - 2x + 4 x^ 2 +  O (x^3) $$
Apply the se a pp roximati ons to yo ur function .You 'lls t ill h avea  f actor of  $1/x^ 2 $in  both t er ms. Asi mpl e p olyno m ialapprox imationdoesn ' t exist.

 Q:How to spawnd et a c he d  com man d in Node i n si l ent/ not_ s il ent mode? 

I'm  m a ki n g automation script  f or mynodeproject and I 've met li t t l e issuewhic h  I can not solve. 
I wa nt to star t  3 detac hed p roccesse s using g r u nt tasks : s ele niu m-s t a nd alone start f o r  tes ts , mongo d - -dbpath ./m ong o a ndno de app.js .
I usesi mi la rcod e for all of t he m
v ar  sp awn = requir e('ch ild_ pr oc e ss' ).spawn , 
c o m mand  = 's elen ium -s tanda lone . cmd ', // o r "mongod " o r  "no de "args =['start']; //  o r ["--dbpa th ",path.j o i n(proces s.cwd() + "/mongo/")] o r  ['app. js' ]
var  ch= spawn(c omm and, a rgs ,  {
            det a c hed:t r ue ,
           en v: pr oc ess. env,
               stdi o 
<?php 
foreach($row_as $r):
?>
<tr><td><?php_echo $r["title"];?></td></tr>
<?php endforeach;?>
</table>
</td>
<?php endforeach;?>
</tr>
</table>

_Q:

Simplify the_following_expression for_small_x

$$ \dfrac{\left(x+1\right)^2}{x^2\left(2x+1\right)}-\dfrac{\left(x+1\right)\ln\left(2x+1\right)}{x^3} $$
How_to simplify the_above expression for small_$x$? I'm having_trouble_with $x^2$ and $x^3$ in the denominators. The other factors, I can Taylor expand_but_what to_do_with_$x^2$ and $x^3$? The end_result should be a simple_expression like_$a+bx+cx^2$. Hints are also fine :)

A:

Hint: For $|x|_\ll_1$, we have
$$_\ln(1 + 2x) \approx 2x - 8x^3 + O(x^5)_$$
$$ \frac{1}{1+2x} \approx 1 - 2x_+ 4x^2 +_O(x^3)_$$
Apply_these approximations to your_function. You'll still have a factor_of $1/x^2$ in both terms. A_simple polynomial approximation doesn't exist.

 Q:

How to_spawn detached command in Node in_silent/not_silent mode?

I'm making automation script_for my_node project and I've met_little issue which_I can_not solve.
I want_to start 3 detached proccesses using_grunt tasks: selenium-standalone_start for tests, mongod --dbpath./mongo and_node_app.js.
I use similar_code_for_all of_them
var spawn =_require('child_process').spawn,
command_='selenium-standalone.cmd', //_or_"mongod" or "node"
args = ['start']; //_or_["--dbpath", path.join(process.cwd() + "/mongo/")] or ['app.js']
var ch_= spawn(command, args, {
__   _    _     detached:_true,
 _ _          _ env: process.env,
   _   _ __  __ stdio