$$S_n = \int_{0}^{1}\frac{n x^{n-1}}{1+x} dx \quad n \ge 1$$  My attempt:  $$|S_n| \le \int_{0}^{1}\left|\frac{n x^{n-1}}{1+x} dx\right| = \int_{0}^{1}\frac{n x^{n-1}}{1+x} dx \le \int_{0}^{1}n x^{n-1} dx$$  $0 \le x \le 1$, and $n \ge 1$.  So, $|S_n|=[x^n]_{0}^{1}$ and thus $|Sn| = 1$.  So, $\lim_{n \to \infty} S_n = 1$.  But the answer mentioned is $0$. I don't know what I am doing wrong. Please help.
Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to \infty} s_n$.   \begin{align}   s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx  \end{align}  I have written a solution of my own, but I would like to know it is completely correct, and also I would like if people post more alternative solutions.
Error to calculate date diff with DateTime class Is it possible to embed a pre-existing DLL into a compiled C# executable (so that you only have one file to distribute)? If it is possible, how would one go about doing it?  Normally, I'm cool with just leaving the DLLs outside and having the setup program handle everything, but there have been a couple of people at work who have asked me this and I honestly don't know. A dollar amount $\${-}67.9{-}$ is divisible by 72. What is this number? A client buys 72 turkeys. On the receipt, 2 digits are missing, the first and the last. So, the total price is $\${-}67.9{-}$. Find a way to make the price of each turkey round at the second digit without trial-and-error method.  So, the answer is obviously $\$5.11$/turkey, so $\$367.92$ total. I found that with with 2-3 trial-and-error.  I bypassed his problem by using python and made a script that tries every combination and assures it is divisible by 0.01.   Please help me find a mathematical way of solving this problem. Honestly, navigation through Area 51 is a bit confusing and seems to be much different from the other SE sites.    The feature that is very lacking is the footer with all the sites you are subscribed to so that a user can easily return to a different SE site. This one really bugs me.  Can this be easily fixed?  Also, unlike all the other sites there is no "meta" site to add comments specifically about that site.    Again Area 51 seems out of sync with the other SE sites. How do you apply a QgsExpression to a Label ?   I can do it in QGIS with this simple expression  concat('N:' ,round(Value,0),'\n','Yield:' ,round(Value 2,0))  N: 65   Yield: 40  How do I do it in PyQGIS ? I see a getLabelExpression but no setLabelExpression  This works but I want but only for the field name    crap = QgsVectorLayer("D:/data.shp", "data", "ogr")  QgsMapLayerRegistry.instance().addMapLayer(crap)  palyr = QgsPalLayerSettings()  palyr.readFromLayer(crap)  palyr.enabled = True  palyr.fieldName = 'Name'  palyr.placement= QgsPalLayerSettings.OverPoint  palyr.setDataDefinedProperty(QgsPalLayerSettings.Size,True,True,'8','')  palyr.writeToLayer(crap) Setting up a bicycle for someone with restricted knee articulation My father, now in his 70's, had a left knee replacement several years ago. He has limited movement in that knee - probably about 85 degrees on a good day. His knee can extend fully, but cannot bend as much as it should. As a result, he cannot pedal a bike he cannot get his left leg over the top of the pedal stroke.   He is desperate to get back onto a bike, he knows he won't be ripping up the road and winning races like the old days, but wants the freedom to go for a slow easy rides.    Does anyone have suggestions as to a solution to the problem. We don't mind spending money if we know the solution will work, but he can't afford to throw big money at wild ideas that may or may not work. If we find a solution, a new bike is likely.   We have access several mountain bikes in a range of sizes to trail ideas, and we have all the tools and skills to build bikes up from parts and rearrange things as needed.    EDIT: Thanks everyone - all great suggestions. What am am going to do is got tto the LBS and get hold of a cheap set of short cranks for around $30- probably from a kids bike, in the 120-150 size range, and see if he can ride that. Depending how he goes, I'll try a standard right crank and a short left crank. If all else fails, we will go for the Power Cranks   UPDATE: In case anyone else has similar problems. Got some 152mm cranks for $30. Set an old  mountain bike up with the seat as far back as possible. The old man is able to ride reasonably comfortably, now has a smile from ear to ear - any suggestions how to remove it - doesn't fit with the grumpy old man image :)   Has got me wondering how many people with minor knee problems would benefit from short(er) cranks....... Can I levitate an object without using an electromagnet? Good-morning Math Exchange (and good evening to some!)  I have a very basic question that is confusing me.   At school I was told that $\sqrt {a^2} = \pm a$  However, does this mean that $\sqrt {a^2} = +2$ *and*$-2$ or does it mean: $\sqrt {a^2} = +2$ *OR*$-2$  Is it wrong to say 'and'? What are the implications of choosing 'and'/'or'  Any help would be greatly appreciated. Thanks in advance and enjoy the rest of your day :) How to get all possible combinations of a list’s elements? Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$ I'm trying to configure a ubuntu server network. [Edit] My ifconfig: lo: flags=73&lt;UP,LOOPBACK,RUNNING&gt; mtu 65536 inet 127.0.0.1 netmask 255.0.0.0 inet6 ::1 prefixlen 128 scopeid 0x10&lt;host&gt; loop txqueuelen 1000 (Loopback Local) RX packets 352 bytes 26160 (26.1KB) RX erors 0 dropped 0 overruns 0 frame 0 TX packets 352 bytes 26160 (26.1KB) TX errors 0 dropped 0 overruns 0 carrier 0 collisions 0  I edited the /etc/network/interfaces file and here is how it is: auto lo auto eth0 iface eth0 inet static address 192.168.15.141 netmask 255.255.255.0 gateway 192.168.15.1  I'm trying to restart the server and I'm using: sudo ifdown --exclude=lo -a &amp;&amp; sudo ifup --exclude=lo -a  This command is an answer from this link:  And I'm getting the following error: Cannot find device &quot;eth0&quot; Failed to bring up eth0.  I'm pretty lay in configuring a server, so I have no idea what's going on. Can someone help me please? (I'm aware of  but it has no answers…)  Let be $u_0 \in \mathbb{R}$ and the sequence $(u_n)_n$ defined by: $u_{n + 1} = \frac12 \arctan(u_n)$.  I define also: $v_n = 2^n u_n$, so I can show that: $\lim (u_n)_n = 0$ (by studying $x \mapsto \frac12 \arctan(x)$), thus, I can show that $(v_n)_n$ is monotone and converges because it is bound.  Now, I conclude: $u_n \sim \dfrac{l}{2^n}$, I'd like to determine $l$ more precisely.  Here is what I tried, I suspect $l$ to be something like $f(\pi)$ for some $f$ :   push the asymptotic expansion of $\arctan$ to the 2nd order and reinject it ; use $\arctan(u_n) + \arctan(1/u_n) = \dfrac{\pi}{2}$ ; use series techniques to look for $\sum v_{n + 1} - v_n$, maybe conclude using Cesaro summation Determine the following limit as x approaches 0: $\frac{\ln(1+x)}x$ $$\lim_{x\to 0} \frac{\ln(1+x)}x$$  The process I  want to take to solving this is by using the definition of the limit, but I am getting confused. ( without l'hopitals rule)   $$\lim_{h \to 0} \frac{f(x+h) - f(x)}h$$  $$\lim_{h \to 0} \frac{\frac{\ln (1+x+h)}{x+h} - \frac{\ln(1+x)}x}h$$  $$\lim_{h \to 0} \frac{x\ln(1+x+h) - (x+h)\ln (1+x)}{hx(x+h))}$$  At this point I get confused because I know the answer is $1$, but I am not getting this answer through simplification of my formula. The HDD I have my Ubuntu installed is about to fail. I would rather not lose 3 years worth of data, customisation and apps. I am looking for a way to move the complete system (SWAP included, because I'm not sure if I can relink the system to a new SWAP partition) to another HDD. But not the complete HDD&lt; only the partition containing Ubuntu, to a partition on a different HDD. Basically I'd like to do what I've been able to do with Norton Ghost for my Windows install. I thought about using Clonezilla but I think I would have issues with GRUB (Especially trying to boot from a different UUID than what is in the conf file). do you know of any way this could be done?  PS, my home directory is encrypted but that's not really an issue, because I can work around that.  EDIT: changed the explanation to make it clearer I'm level 16 which makes battling many Draugr Wrights hard. My defense is just way too low, and I don't have enough time or strong enough magic to take them out.   My one-handed skill is at 62 and my strongest weapon does 33+10 enchanted damage. Are there any ways to level up easily? I would like to find a Python script that will access an attribute table within a shapefile and either update or replace records inside a field column. More specifically, I would like to use the replace expression to convert a table record like "123-456-789" to "123456789". I have many records to process and would like the best approach to do this task using Python. Getting video dimension / resolution / width x height from ffmpeg File changed listener in Java Best way to start investing, for a young person just starting their career? I am using Paint to draw some swords for a character I have. I am not very experienced and this is my first time trying to draw swords.  My question is how large (Preferably in pixels) do I need to make the picture to make the curves on the sword to look smooth. I am using current scale of 100 pixels = an inch. For the first sword I made this worked fine as there were few curves and they were small but the one I am working on now have a larger curve. (A good reference is .)   EDIT: I only have access to paint. I would use other programs if I could.