The question is basically to prove that convergence of $\sum{a_n}$  implies convergence of $\sum {\frac{\sqrt a_n}n}$  with $a_n&gt;0.$  Basically what I tried to show is that:  Since $\sum{a_n}$ converges, then $\lim a_n= 0$ when $n \rightarrow \infty$ , and so $\sqrt a_n \rightarrow 1$ when $n \rightarrow \infty,$ which means it is bounded. On the other hand, $\frac1n \rightarrow 0$ as $n \rightarrow \infty$ and is always decreasing. Per theorem 3.42 in Rudin, that means $\sum{\frac{\sqrt a_n}n}$ also converges.  Is that correct or am I missing something? The solutions on the net appear to differ.
This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \ge 0$.  Proof:  I will attempt to show that the cauchy criterion is met so $${\sum_{k=n}^m {\sqrt {a_k} \over k}} \le \epsilon $$ for all $n,m \le N_1$. Im disregarding the absolute value since its all positive terms.   Since $\sum{a_n}$ converges, we know that the $\lim \limits_{n \to \infty} a_n =0$ So there exists some $N_2$ such that for all $n \ge N_2$, $$   a_n \lt \ {{(m-n)}^2k^2(\epsilon)^2} $$ I am again disregarding the absolute values because its always positive Let N=max{$N_1,N_2$} then for all $n,m \ge N$ $$\sum_{k=n}^m {\sqrt {a_k} \over k} \le \sum_{k=n}^m {\sqrt {{(m-n)}^2k^2(\epsilon)^2} \over k} \le \sum_{k=n}^m{(m-n)(\epsilon)} = \epsilon $$  Maybe this is just crazy, should I just use Cauchy-Schwarz inequality and forget about this or could I possibly make this method work?
I'm giving Deus Ex a quick playthrough as a refresher before jumping into Human Revolution. In the interest of expediency, I've decided to cheat. ^.^  Normally, cheats must be re-enabled for each level when playing the original DX. I've encountered instructions on several sites (word-for-word identical instructions, so clearly copy/pasted) for permanently enabling cheats, but they aren't working for me:     Because you will need to redo this for every level, you can open up 'user.ini' and add 'bCheatsEnabled True' in the [DeusEx.DeusExGameInfo] section of the file to allow cheats to be activated forever.   Not only is that not working for me, it doesn't even look right compared to the rest of the file (space instead of =). With increasing desperation, I've tried adding that line (both verbatim and switching to an equal sign) to every section of both User.ini and DeusEx.ini which looks even remotely plausible — even creating a [DeusEx.JCDentonMale] section, just in case.  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I know I can work around this by prepending a ' or changing the cell format, but I'm interested in changing the default format.  An approach that almost works is going to Control Panel->Region and Language->Long Time format and changing it to something like "h". This results in Excel displaying the dates I paste in as "2012-12-07 00:00:00.000", but screws up times displayed outside of Excel.  The dates I'm pasting in are from SQL Server. ESTA approved on the same day as B-1/B-2 visa refused I am writing because I have a question regarding my ESTA application and because I could not find a similar case online that would help me at this stage. I have never been to the U.S. so this would be my first visit.    I am a German citizen, hence eligible for the Waiver Program. I applied for an ESTA in June 2014 and got approved. After further research on the Internet I learned that the passport should be valid until 6 months after the planned return date. My passport was due to run out in Feb 2015 - which still should have been fine but because I didn't want to take chances I applied for a new passport which was issued August 2014 and is now valid until August 2024. My passport number has been changed also.    After I received my new passport I was advised by a friend to apply for a B-1/B-2 visa as it would allow me to extend my stay and it seemed like a good idea at first. On the day of my interview I also applied for a new ESTA with my new passport number (just in case) which again was granted.   Unfortunately for me the interview for the B-1/B-2 visa was unsuccessful and I got rejected. I was not aware then that after a rejected interview one could not travel with ESTA, otherwise I would have never risked it!  Now I am very concerned, can I still use my ESTA and enter the U.S.? as I read everywhere that once you get rejected the chances of getting through are very slim and any visa rejection will result in not being able to use the waiver program.  It might be important to note that both the B-1/B-2 interview and the waiver application were on the same day, 21st of August 2014, so depending on how one might look at it that might present a problem too I suppose.  I guess the frustration lies in the fact that I did not need to apply for a B-1/B-2 visa to begin with, and had I not I would now not have these problems. I would like to know how likely it is that I can still travel under the waiver program, if at all, and what my options are at this point. One of my friends argues that princessship is the only word which has 3 identical letter comes together (s) ,but I think there is no word such as princessship.  Can anyone tell me whether this is a real word, and if there are any more English words which have 3 letters consecutively? font set problems in beamer i want to set the main font in my beamer as "Consolas" i do like this :  \documentclass{beamer} \usepackage{fontspec} \usetheme{Warsaw} \begin{document} \setmainfont{Consolas} \begin{frame} we discuss the ambiguities of existing security definitions and then provide a general framework for defining AKE security when involve strong adversary capabilities \end{frame} \end{document}   but after compile it , the font doesn't change, it is the same with that before the main font was set. what's the reason? Solving an equation over the reals: $ x^3 + 1 = 2\sqrt[3]{{2x - 1}}$ Solve the following equation over the reals:$$ x^3  + 1 = 2\sqrt[3]{{2x - 1}} $$ I noticed that 1 is a trivial solution, then I tried raising the equation to the 3rd, then dividing the polynomial by $(x-1)$.. But I can't see the solution, how do I go from here? How to open a new Terminal with shell prompt and running multiple commands in Command Line? I want to Open a new Terminal in from within terminal.  It should have Shell prompt as well as already run few commands.  How to do it ? A question about eigenvalues of $5\times5$ matrix $A\in M_5({R})$  and $A^2-4A-I=0$ If $a_1,a_2,a_3,a_4,a_5$ are eigenvalues of $A$ what is value of $$\left(a_1-\dfrac{1}{a_1}\right)+\left(a_2-\dfrac{1}{a_2}\right)+\cdots+\left(a_5-\dfrac{1}{a_5}\right)$$ I tried to work with $A^2-4A-I=0$ but get nothing because $A_{5\times5}$ characteristic polynomial is something like $A^5+k_1A^4+...$ Any hint will be appreciated . I accidentally installed some USB drivers to my Windows 7 system, they were the wrong ones and now my keyboard and mouse won't work. A system restore would get rid of them, but obviously I can't control my computer to make that happen.  Keyboard and mouse still work before windows has loaded, in the BIOS and boot system, so can I do anything from there? I've already tried booting into another OS, accessing my windows filesystem and deleting the Program Files entry, but it didnt work so they must go deeper than that. Or could i launch it from a remote machine? I've got some VNC server insalled so if i could get that running that could work. Does metric signature affect the stress energy tensor? If one were to derive the stress-energy tensor for a metric with $(+,-,-,-)$ signature would it be different from the stress-energy tensor derived from the same metric but with $(-,+,+,+)$ signature? I don't mean people with low self esteem.  But there are people who feel happy at being able to make other people laugh at their own expense. I remember someone telling me there is a word for them, I can't find it.  Let me explain:  I sometimes, while sitting with a group of people, might just tell them a personal embarrassing story, which might be completely fabricated, to make other people laugh. And when they laugh, I feel happiness.  I'm also self deprecating, but not in a way that I hate myself, rather,  just my work (I'm a screenwriter) and knowing I can do better. I do not self deprecate myself in public (I don't praise myself either).   Edit: Now that its mentioned, I think the word I'm talking about was used to describe a person's sense of humor. Or it could be a word used to describe a psychological trait/disorder.  Edit: I just looked up humor styles on Wikipedia. And after reading it, I guess the word that closely matches what I'm looking for is 'Affiliated Humor' but in a 'Self Defeating' style. I am trying to render a kitchen sink / Faucet    Here is another reference image of what I want   What I have got so far is this   Settings for World:  I have used an environment map HDRI with 0 saturation.  Material settings:  Triad with glossy, glass anisotropic and there mix shader combinations and the new principled bsdf shader too but didn't get the required result.  Studio lights: Followed multiple youtube tutorials on setting up studio lights in blender but the results were not what I wanted.    LEAVING CYCLES :  Decided to try some other software So, Downloaded Keyshot 5 but keyshot wasn't able to produce cycles like results LoL.  Not : I have always struggled to get professional quality product rendering in cycles      So now I'm back in cycles looking for help. How to hide quoted text in Lion Mail? In Mail's conversation view, by default, quoted text is hidden and can be shown by clicking "See More from XXX" in the bottom of each message. After clicking that, how can quoted text be hidden again? If $x\sqrt{1+y}+y\sqrt{1+x}=0$ find $y'$ I want to run Notepad++ twice. Like in this paint printscreen.  How I want it:    Not what I want:    Is this possible? I wasn't forcing myself to say anything, or/nor trying to be funny.   "Or" sounds more grammatically correct to me. But "nor" has more Google results. So I'm confused. Why does Loki look "sick" when he first arrives on Earth? In the Avengers movie, why is Loki sweating and weak when he first arrives on Earth, in the SHIELD base?