Show that $1/\sqrt{1} + 1/\sqrt{2} + ... + 1/\sqrt{n} \leq 2\sqrt{n}-1$ for $n\geq 1$  I attempted the problem but I get stuck trying to show that if the statment is true for some $k\geq1$ then $k+1$ is also true.  My attempt: Base case $n=1$ is true by inspection  Now assume the statement is true for some $k\geq1$. Then $1/\sqrt{1} + 1/\sqrt{2} + ... + 1/\sqrt{n} +1\sqrt{n+1} \leq 2\sqrt{n}-1 + 1/\sqrt{n+1}$ but I am not sure how to show the rest.
Prove that for any positive integer $n$, $$2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$$ Progress I think Riemann sum should be used for the middle term. I got the limit as $n\to \infty$ of the function inside using Riemann sum but what I cannot get are the lower and upper bound.
Get string after character Is it allowed/desirable to post code snippets? Instead of asking a question I would love to share the one or other code snippet I have created that is too small to make a full article on e.g. CodeProject.com or CodePlex.com.  Question: Would it be a good idea to post it as a Stack Overflow question? Or are there any other suggestions how to share it?  (Of course I could post it on my own weblog and then twitter about it, but I would love to put it in more of a programming context) String concatenation: concat() vs "+" operator Assuming String a and b:  a += b a = a.concat(b)   Under the hood, are they the same thing?  Here is concat decompiled as reference. I'd like to be able to decompile the + operator as well to see what that does.  public String concat(String s) {      int i = s.length();     if (i == 0) {         return this;     }     else {         char ac[] = new char[count + i];         getChars(0, count, ac, 0);         s.getChars(0, i, ac, count);         return new String(0, count + i, ac);     } } Do greenhouse gasses make the world habitable? An ageing gent says he does not accept this statement:  It's a fact that greenhouse gases, predominantly water vapour, keep the Earth  from being hostile and cold; that CO2 albeit in trace quantities is also a  greenhouse gas and therefore helps keep the planet warm...   I know no more about why he rejects this statement - maybe it is that water vapour alone would do the job. The gent in question has a first class degree and a doctorate in physics.    Is there a debate within the physics community over this or am I misunderstanding? What do we lose passing from the reals to the complex numbers? As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we lose moving from the reals to the complex numbers? Content type permission to specific users How do I set content permission so that specific content types become visible/usable to the admin user only, while other content types may be visible by other user roles?  After I added new content types, I checked the rebuild permissions.   Content access is working fine but the problem seems to be the fact that the enw content types aren't made visible / accessible to the other roles... How to switch from Natbib to Biblatex in pre-made document class/template? I am using  to write my thesis. The template uses Natlib for references, whereas I really need Biblatex.   When I simply try to change \usepackage[natbib] for biblatex I get the error message  This is BibTeX, Version 0.99d (TeX Live 2016/Debian) Capacity: max_strings=100000, hash_size=100000, hash_prime=85009 The top-level auxiliary file: aagot-1-report.aux The style file: acm.bst Illegal, another \bibstyle command---line 80 of file aagot-1-report.aux  : \bibstyle  :          {acm}  I'm skipping whatever remains of this command Database file #1: Mendeleytry.bib.bib Warning--I didn't find a database entry for "" You've used 14 entries,code here   The solution  \usepackage[backend=bibtex, natbib=true, style=numeric]{biblatex}   didn't work.  Another possible solution was said to be to recompile from scratch to delete the .aux file, but that resulted in  /usr/share/texlive/texmf-dist/tex/latex/biblatex/biblatex.sty:462: LaTeX Error:  Command \bibhang already defined.            Or name \end... illegal, see p.192 of the manual.  See the LaTeX manual or LaTeX Companion for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.462 \newlength{\bibhang}   I have tried to use this "marco" to override the \requirePackage settings, but that also didn't work and resulted into  LaTeX Error: Two \documentclass or \documentstyle comm ands.  See the LaTeX manual or LaTeX Companion for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.45 \documentclass[whitelogo]{                            tudelft-report}    Where I can't seem to delete any of the document classes without getting the functionality of the tudelft-package and the biblatex citing (or in fact getting it to work at all!)  (no line number): [0] Config.pm:343&gt; INFO - This is Biber 2.5 [0] Config.pm:346&gt; INFO - Logfile is 'aagot-1-report.blg' [62] biber:290&gt; INFO - === Tue Apr 24, 2018, 22:24:06 [246] Utils.pm:165&gt; ERROR - aagot-1-report.bcf is malformed, last biblatex  run probably failed. Deleted aagot-1-report.bbl [246] Biber.pm:113&gt; INFO - ERRORS: 1   Is there any way I can still make this template work with Biblatex? Natlib seems to deeply rooted in the file and I don't know how to change it/access the settings/the ".sty"-file. Can I use the example documentclass combined with biblatex (instead of the standard natbib) without losing the great parts of the rest of the document class?  Thanks very much :) Python: using a recursive algorithm as a generator How do I create a new object using Python in Blender 2.80? As part of an existing add-on, I'm trying to create an object and add it to the scene using Python. I followed a Blender Cookbook tutorial found here: . I use the following code to do so:  emptyMesh = bpy.data.meshes.new('emptyMesh') theObj = bpy.data.objects.new("object_name", emptyMesh) bpy.context.scene.objects.link(theObj)   But I get the following error when I try to link theObj to the scene objects:  AttributeError: 'bpy_prop_collection' object has no attribute 'link'   This works just fine in Blender 2.76-2.79, but doesn't work in the 2.80 test build I downloaded. Was there a change to the Python API? Or is this maybe a bug in the test build? Does Google use the Meta Description Tag for Description of Page? Leave bash script running on remote terminal while not logged in? I have a bash script that takes several hours to run.  While it's running, I would like to do other things, which may involve logging out or disconnecting from the internet (my script runs network tests on various computers).  I understand that there is a command that would allow me to run my tests from a remote terminal and logout of the terminal while it runs.  Does anyone know what this command is? How do users get started on new Stack Exchange sites? How do users on the new Stack Exchange sites get started? Each user starts with 1 rep, and can't upvote until they have 15 rep.  It struck me that this creates a bit of a deadlock. If a new user asks a question and a new user answers, neither can upvote the other.  Are we relying solely on existing SE users to join these exchange sites (with their bonus 100 rep) so that they can get started? Is it possible to install Python 3.x in 14.04 LTS, or upgrade from Python 2.7.6 to 2.7.8? When I capture packets using the packet analyser wireshark while visiting  I can see packets that use the HTTP protocol (e.g.a GET request). My Get request uses 4 protocols: Ethernet II, IPv4, TCP and HTTP.  For the IPv4 I can see my own local IPv4 address of my pc as the source, for the destination I see the IPv4 address 91.198.174.225. When I go to this address in my browser I get to some wikipedia related website that says'wiki does not exist'. Why am I not seeing the Hydrogen article when I go to that ip address? Proving $\sum_{k=1}^n k k!=(n+1)!-1$ On any SE post one can use the backtick symbol ("`") in order to mark some text as being a piece of code, or computer output. For example: void main().  Is there a way to make use of "`" the same way in a LaTeX document? How to install Unity 8? I'm using Ubuntu 14.04 and found out that we can install Unity 8 on it. Will Unity 8 replace Unity 7 once I install it? And if it does can I switch back to Unity 7 if I want? How would one go about it? How to create two classes in C++ which use each other as data? Some months ago I watched  video of Blenderguru about "How to create a realistic earth in blender". In this video blender render is used and  now that I understand texturing in cycles a little better I am trying to rebuild this "realistic earth" with cycles.    In the video - starting at about 11:30 min - it is shown how to map the "earth day" and "earth night" image textures. Depending on the lights shining on the earth the earth or night image is projected. I tried to rebuild that in cycles but I think I a making mistakes. In the circled box the "night" nodes are found. What Is my mistake ?   P.S When you see other mistakes in this node tree, please let me know.  OK ... I now have implemented the advice of Uncle Snail (except moving the black slider). Below an image of the earth and the updated node tree (I also rearranged nodes for better overview).    In the Color Ramp, I did not move the slider. Is that really necessary ?? I wanted to know if there is difference in terms of charging the battery when a MacBook Pro 16" is connected to two monitors via USB-C, both with Power Delivery capability.   In this specific case I am talking about the LG 27UK850-W monitors. The MacBook Pro 16" will be connected via USB-C to both of them and use them as external monitors. Since both can charge it as well, will the MacBook Pro  charge faster while being connected to both or will just one monitor be able to deliver the charge at a time?