Let $s_n=1+\frac{1}{\sqrt2}+\cdots +\frac{1}{\sqrt n}$. This sequence is definitely increasing, and I tried to show that it is not bounded above, by induction. However if $s_n\leq2$, then $s_{n+1}\leq2+\frac{1}{\sqrt n}\nleq2$. So I thought this is not bounded above.  Now, the second sequence is $s_n=\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2}$. The problem requires to prove that this one is bounded above by $2$.  I tried to apply the same method as before: if $s_n\leq2$, then $s_{n+1}\leq2+\frac{1}{(n+1)^2}\nleq2$. But, this is bounded above by $2$.  What is wrong with my logic?
$$\dfrac{1}{\sqrt 1}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\cdots+\dfrac{1}{\sqrt n}\geq \sqrt n$$ I want to prove this by Induction  $$n=1 \checkmark\\ n=k \to \dfrac{1}{\sqrt 1}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\cdots+\dfrac{1}{\sqrt k}\geq \sqrt k\\ n=k+1 \to \dfrac{1}{\sqrt 1}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\cdots+\dfrac{1}{\sqrt {k+1}}\geq \sqrt {k+1}$$ so $$\dfrac{1}{\sqrt 1}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\cdots+\dfrac{1}{\sqrt k}+\dfrac{1}{\sqrt {k+1}}\geq \sqrt k+\dfrac{1}{\sqrt {k+1}}$$now we prove that $$\sqrt k+\dfrac{1}{\sqrt {k+1}} &gt;\sqrt{k+1} \\\sqrt{k(k+1)}+1 \geq k+1 \\ k(k+1) \geq k^2 \\k+1 \geq k \checkmark$$ and the second method like below ,   and I want to know is there more Idia to show this proof  ? forexample  combinatorics proofs  , or using integrals ,or fourier series ,....    Is there a close form for this summation ?  any help will be appreciated .
Uninstall kubuntu-desktop I installed kubuntu-desktop using sudo apt-get install kubuntu-desktop but I no longer want to use KDE.  How do I remove all the packages that were installed by this command ?  Thank you Restore a SQL Server 2012 backup file in SQL Server 2016/2014 I have SQL Server 2012 backup files and want to restore these to SQL Server 2016/2014. Will it work?  I need to find this out before I uninstall SQL Server 2012. This question is prompted by:  and .  Basically, some diamond moderators don't always want to diamond-hammer close questions.  A workaround is to create a new account, , then cast regular close votes from that account.  Question: May a diamond moderator utilize a second, non-diamond account e.g. to cast regular close votes?     It is not a problem to have multiple accounts, as long as they aren't voting for each other and doing other sockpuppet-y things.      I worry this might be considered "sockpuppet-y".  The diamond close hammer is sometimes excessive, particularly  where moderators are . The strange thing is that opened scripts or newly opened ones are working fine. It's the new created C# scripts that are not working.  With not working, I mean the MonoBehaviour is not in light blue color it's in black color. In this screenshot I took, I marked it with a red circle.    What have I tried so far? I have closed Visual Studio and started over again.  Since it didn't solve it, I closed the unity editor and started it over again, to no avail.  I pressed 'clean solution', I pressed 'rebuild solution', I pressed 'build solution'.  What else can I do ? And why are the other scripts working fine? If I type gameo...it will auto complete it to GameObject but in this specific script it's just not working.  This is a screenshot of the editor. I have in the Hierarchy a Ladder object.  In the Assets I created a folder name Ladder and a script name Ladder. But this Ladder script is not working.    This is the Ladder script and this script was working before but once I created another script inside the ladder folder name Raise empty script the Raise script didn't work and now the Ladder is not working either. But other scripts are working fine.  using System.Collections; using System.Collections.Generic; using UnityEngine;  public class Ladder : MonoBehaviour {     public Transform charcontroller;      private bool inside = false;     private float heightFactor = 3.2f;      private void OnTriggerEnter(Collider other)     {             inside = true;     }      private void OnTriggerExit(Collider other)     {             inside = false;     }      private void Update()     {         if (inside == true &amp;&amp; Input.GetKey("w"))             charcontroller.transform.position += Vector3.up / heightFactor;     } } How to read the following Itinerary What do the abbreviations mean in the following itinerary:     2  EK 641 L 29NOV 2 KBLDXB HK1       I  1630 1905   *1A/E*                      3  EK 344 L 29NOV 2 DXBKUL HK1       3  2140 0835+1 *1A/E*                      4  MH 704 V 30NOV 3 KULMNL HK1       M  1245 1630   *1A/E*                      5  UA 184 Y 01DEC 4 MNLGUM HK1       1  2255 0440+1 *1A/E*                      6  UA 155 Y 02DEC 5 GUMTKK HK1          0820 1008   especially, the numbers (2,2,3,4,5) before KBLDXB, KULMNL.... The letters and numbers before time (I, 3, M, 1) *1A/E*? Is it appropriate to use the salutation "Dear All" in a work email? I have observed that in my work place, whenever a mail is sent to more than one person( like an information, meeting request or a notice etc.), the mail starts with the salutation "Dear All". This, somehow, doesn't seem correct to me. For me, a salutation should address the reader individually, at least in its singular form, as in "Dear Colleagues".  Am I correct in thinking that? How do you Set Custom Time in Ubuntu? How to use more than 10 tab stops in bmatrix or other amsmath matrix environments? I'm using bmatrix environment from the amsmath package, and I'm trying to do something like this:  \documentclass{article} \usepackage{amsmath}   \begin{document}      \begin{equation}          \begin{bmatrix}              1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 &amp; 9 &amp; 10 &amp; 11\\          \end{bmatrix}     \end{equation}  \end{document}   However, pdfLaTeX returns an error: "Extra alignment tab has been changed to \cr", and in the output the extra element is moved to the next row. (If I remove one element, then it works perfectly fine). I guess this means that I've hit some hard limit on the number of tab stops in a matrix. Is there a way to change this?  The elements in the matrix are short and could fit perfectly on a single row. Is there a difference between “?!” and “!?”? This is perhaps a simple question with a simple answer, but I'm stumped as a tikz novice. What would be the best method to add an image (logo) file to the upper left-hand corner of my document?    MWE:  \documentclass[12pt]{article}  \usepackage{lipsum} \usepackage{showframe} %\usepackage{tikz} %\usepackage{graphicx}  \begin{document}  \section*{Lorem Ipsum}  \lipsum[1]  \end{document}   I wish to avoid using fancyhdr to achieve this result. How do I know if an ability is magical? Is there a closed formula for this summation? I have the summation  $$\sum\limits_{i=1}^n \frac{1}{i^2}$$  And I don't know how to find a closed formula for it. Any ideas? I predicted the formula by finding total number of non-isomorphic labelled trees of order 1 is 1 , order 2 is 1,order 3 is 3,order 4 is 16,order 5 is 125.But how do i prove it ? I am beginner in graph theory so i will be very thankful if i get some simple approach to solve the problem(Not by using some difficult theorems). $N_1,N_2,N_3 \unlhd G, N_i\cap N_j =\{e\}, G = N_iN_j$. Want to show that $G$ is abelian, $N_i$ are isomorphic. The following is a problem from the Berkeley Problems book.      Let $G$ be a group with three normal subgroups $N_1$ , $N_2$, and   $N_3$. Suppose $N_i \cap N_j = \{ e\}$ and $N_iN_j  = G$ for all   $(i,j )$ with $i \ne j$ . Show that $G$ is abelian and $N_i$, is   isomorphic to $N_j$ for all $i, j$ .   Attempt:   I have not got very far. The only thing that comes to my mind is that as $|HK|=\frac{|H||K|}{|H \cap K|}$, $|N_1|=|N_2|=|N_3|=m$, and order of $|G|=m^2$. I have made other futile attempts of trying to show m prime, finding an isomorphism between the $N's$. I think the fact that the equation $N_i N_j=G$, could be seen as, left coset of $N_j$ with the multiplying elements restricted to only $N_i$ could provide me an answer, but I haven't got anywhere with this.   Thanks in advance. We have a view for a type of node that includes a Private File.  We cannot see a way to add an exposed filter, such as "Contains any word", and have it filter based on the file's name.  Are we missing something obvious? Can we have a system to submit potential robo-reviewers for moderators attention? Find the limit: $\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$ Introductory text for lattice theory I've watched the Vampire Knight anime. I really like the story so far but find it frustrating that the anime doesn't finish off the story. I'm trying to figure out if it is worth reading the manga.  Does the manga contain more story than the anime? Does it finish the story off? Failed to save undefined: No COMPONENT named markup://svg found :