If $\sum a^2_n$ converges so does $\sum \frac{a_n}{n}$   I'm trying to prove/ disprove this statement. But I couldn't end up with anything yet. Any help?
Suppose $\sum{a_n}$ converges. How do I prove that $\sum{\frac{\sqrt{a_n}}{n}}$ converges.  So I know that just because  $\sum{a_n}$ converges, do not mean I can say anything about the converges of its square root. So I know that I can prove that if  $\sum{\sqrt{a_n}}$ converges then $\sum{\frac{\sqrt{a_n}}{n}}$ converges by Abel's Test, but I do not know where to start for the case where  $\sum{\frac{\sqrt{a_n}}{n}}$ diverges
I know cell phone cameras are the lowest rung, but when reading reviews of phones, the reviewers always have sample shots that are much better than what I can manage.  What are some ways to take decent shots with a cell phone camera?  (I'm thinking of the higher end Android or iPhone cameras). Prove that $f(x)$ is irreducible iff its reciprocal polynomial $f^*(x)$ is irreducible. This is what I'm trying to prove:     Let $f(x)\in\mathbb{Q}[x]$ and $\deg(f(x))&gt;1$. Prove that $f(x)$ is irreducible in $\mathbb{Q}[x]$ iff its reciprocal polynomial $f^*(x)$ is irreducible in $\mathbb{Q}[x]$.    Note: The reciprocal polynomial $f^*(x)=x^nf(1/x) \in \mathbb{Q}[x]$, where $n=\deg f$.   So my thoughts are to prove the contrapositive, i.e. $f^*(x)$ is reducible iff $f(x)$ is reducible.   So to prove the ($\Rightarrow$) direction I assume that $f^*(x)=g(x)h(x)$, so I get that $g(x)h(x)=x^nf(1/x)$ which implies $f(1/x)=(1/x^n)g(x)h(x)$. I want to say this somehow makes $f(x)$ reducible but I am unable to proceed as $f(1/x)\notin \mathbb{Q}[x] $. I thought about substituting $y=1/x$, which gives $f(y)=y^ng(1/y)h(1/y)$, but I am unable to show that this is irreducible in $\mathbb{Q}[y]$. If I could, then it would be irreducible in $\mathbb{Q}[x]$ since $\mathbb{Q}[y]\cong \mathbb{Q}[x]$   Any suggestions or hints will be appreciated. Prove that $4\sin\frac{2\pi}{7}- \tan\frac{\pi}{7}= \sqrt{7}$ . Prove that $$4\sin\frac{2\pi}{7}- \tan\frac{\pi}{7}= \sqrt{7}$$   I think midway calculations are not easy formulas, I can't find what kind of formula transformation to solve! I think the best solution here is using right triangle . . . I have one solution too, but not pretty . ! How do you sort a dictionary by value? I often have to sort a dictionary, consisting of keys &amp; values, by value. For example, I have a hash of words and respective frequencies, that I want to order by frequency.  There is a SortedList which is good for a single value (say frequency), that I want to map it back to the word.   orders by key, not value. Some resort to a , but is there a cleaner way? How to make an alias permanent? I have created an alias:  alias shh='sqlplus hfdora/hfdora@hfd1"     After creating this alias I was able to enter my database only by typing shh.  But after closing my shell, I wasn't able to find the alias next time. Even after typing only alias, shh was not showing in the list.  Is there any file to make an alias permanent so that it will not be erased? Custom entity datetime field in calendar I am on SharePoint 2013 and do not have access to Designer, as a preface.  I am trying to create a list view that filters items by whether or not they were added in the last hour.  I had done this earlier, only with items that were added in the current week, by creating two columns, 'Week Start' and 'Week End', and filtering by Created falling between those two.  Simple enough.  However, filtering list views does not allow for complex functions, nor does the [Today] function recognize time, causing an issue.  Is there any workaround for this?  Thanks. My web application generates a lot of reports in formats like excel sheet, Csv(comma-separated values), fixed width columns, pdf format, html reports etc based on the parameters I select. There is a lot of data in the reports.  Are there any essential criteria for testing report generation that I may not have considered.  I know to test these things:   Data is correct Filters are used Output format is correct   What else is there? If a motherboard doesn't explicitly support a RAM frequency, is still compatible? For example, my  says this:   Supports DDR3 2133(OC)/1866(OC)/1600/1333/1066 non-ECC, un-buffered memory   Is RAM clocked at 2400 MHz compatible? Will the performance be downgraded or it just won't work?  I found other questions but they were related to the CPU cap at 1600 MHz. This cap is related to the motherboard, specs says 2133(OC) max. Make Excel open some hyperlinks with non-default browser A bunch of our network/server/storage hardware uses web interfaces for their administrative portals. I'm currently using an Excel 2010 worksheet to keep track of each device. I have one column set aside to list the URL for each web interface. This makes it easy to simply click the URL to open each administrative portal.  My default browser is Firefox. However, some of the administrative portals don't play well with Firefox and require the use of IE. Until now, when I want to open a hyperlink in IE, I have to manually open IE, copy the URL from the Excel worksheet, and paste it into the IE address bar (otherwise it will open in Firefox if I simply click it). It's just an annoying extra couple of steps.  So without changing Firefox as my default browser, is there a simple way in Excel 2010 to tell it which hyperlinks to open in IE when they are clicked? I want to choose specifically which links open in which browser  (For reference I'm using Windows 7 Enterprise 64-bit) I dont quite understand how pipes work in bash.  I know that it takes an output from one command as the input in another command.  What an output is I can get because it's what the command prints out to the screen.  But how do I know what input a command will take?  Here is an example I thought would work:  which gem | rm   Unfortunately it didn't.  Which gem prints out /usr/bin/gem so that must be the output right?  I thought that was given to rm so it woud be rm /usr/bin/gem but I was wrong.  So my question is, how do I know what input a command takes? If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$ an ideal, then $M$ has a structure of $R/I$-module Let $M$ be an $R$-module and let $I$ be an ideal of $R$ such that $I$ is a subset of $\mathrm{Ann}(M)$. Define a product of an element of $R/I$ by an element of $M$ as follows: $$(r+I)\cdot m=rm.$$ (a) Show that this product is well defined. (b) Show that this product, together with the sum of elements of $M$, turns $M$ into an $R/I$-module.  My answer for (a): Let $r+I,s+I \in R/I$ such that $r+I=s+I$. Then $(r+I)m=(s+I)m$, for all $m \in M$. Then $rm=sm \implies r=s$  (since the inverse of $m$ is in $M$), so the product is well defined.  Is my answer to (a) correct?  How can I use Ann(M) to answer (b)? Is it considered rude if you only ask Questions but almost never answer one yourself I have used the websites of SO a lot the past few months because I am busy with a large scale web project. I feel I get better answers here on SO than when I just do a Google search, maybe because I can be more specific here with my problem, and the responses are really fast.  Occasionally when I am not working I try to answer some questions, but the question almost always has an answer which is 'as good as' or 'better' than the answer I would supply.  So my question/answer ratio is about 20/1 something, if not more. When I look at my profile page I see a bucket-load of questions and almost nothing I answered myself.  Is this considered bad behaviour? I can see that this looks a little egoistic.  : One devoted to one's own interests and advancement. Why do DNGs only show in low resolution on Google Photos? Through my NAS at home I store and backup all my photos on Google Drive. These photos also show up on Google Photos. I use Lightroom and convert all photos to DNG when copying to my NAS.  When I look at the photos in Google Photos the image is rather small. Clicking the (i) info button tells me the image has a resolution of 1024x680, file size 9.8 MB and a MP of 0.7. This is probably just a preview or thumbnail of some sort, but how do I view the image in larger resolutions in Google Photos? $f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ I came across the following problem that says:  Let $f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|&lt;1\}$ such that the range of $f$ is contained in the set $\mathbb{C}\setminus (-\infty,0]$. Then  $f$ is necessarily a constant function. there exists an analytic function $g$ on $\mathbb{D}$ such that $g(x)$ is a square root of $f(z)$ for each $z\in\mathbb{D}$. there exists an analytic function $g$ on $\mathbb{D}$ such that $\operatorname{Re}g(z)\geq 0$ and $g(z)$ is a square root of $f(z)$ for each $z\in\mathbb{D}$. there exists an analytic function $g$ on $\mathbb{D}$ such that $\operatorname{Re}g(z)\leq 0$ and $g(z)$ is a square root of $f(z)$ for each $z\in\mathbb{D}$.   I have to determine which options are correct. Can someone help in the right direction? Thanks in advance for your time. Arduino-controlled slot cars - alternative to MOSFETs for speed control? I'm attempting to control the speed of two slot cars in a set using Arduino.    Each car is controlled by a controller with a variable resistor that looks like this:    They're wired in parallel to the same 17V DC power supply:    Simplified:    For Arduino control, I experimented with manipulating a single car's speed using a MOSFET. It worked really well. Using a PWM output on the Arduino, I was able to pulse the MOSFET on and off to throttle the current and change the car's speed. The behavior of the MOSFET seemed to be exactly the same as the variable resistor controller; the resistance, voltage, and current fluctuated the same way on various parts of the circuit.  Enter car B. I added a MOSFET to the setup and mimicked the MOSFET wiring for car A, grounding them both to the Arduino ground. The result looks like this:    Here's a sketch:    Close up, MOSFET A:    MOSFET B:    At first, all seemed well. I sent signals to MOSFET A and successfully changed Car A's speed. I stopped Car A and pulsed MOSFET B, and I was able to control Car B's speed just fine.  Then, I turned Car B to a constant speed, and sent a signal to MOSFET A while Car B was still running. Car A started, but Car B's speed immediately dropped as a result of running Car A.  I hooked a multimeter up across the rails of Car B and watched its voltage while it ran on its own. Then, sure enough, when Car A started running alongside it, Car B's voltage dropped dramatically.  I figured this was because of my wiring configuration, so I left the multimeter on Car B, and made one change. I replaced MOSFET A with the original slot car controller:    I started running Car B again using its MOSFET. The multimeter showed a steady voltage, so I began to squeeze Car A's controller (the variable resistor), and Car A's speed increased. This time, however, unlike using a MOSFET to control Car A, the variable resistor did not affect Car B's voltage at all. Car B remained at a steady speed the whole time.  Any idea why this happens? The MOSFET seemed to function the same as the variable resistor (empirically) with a single car, but had totally different behavior with two cars in parallel. Is it because the MOSFET is not really changing resistance to current, but just switching it on and off, so it opens up another path for current to flow fully every time it closes? Is there an Arduino-controllable alternative to a MOSFET that would fluctuate actual resistance like the variable resistor? Digital potentiometer? Servo hooked up to a potentiometer? ;) I'm really curious what the difference is and what causes the major differences in behavior. Why does my document 'error' when using '%' I am designing gamecards in LaTeX  Everything is going pretty ok so far but when i try to insert the %-sign for a question about percentages my file keeps giving errors.  I tried putting the percentage between dollarsigns because i know just % will make it a 'remark' and that the dollarsign makes it 'mathematical data'.   Whats the reason for the error?   P.s. the error says: file ended while scanning use of \kaart. ...    Codesample:   {\kaart{Bekijk statistiek kaart F:   Schat het percentage mensen dat als lievelingseten spaghetti opgaven, je mag er slechts $ 2% $ langs zitten. } { $ 11% $}} Can I install items on Ubuntu, without using the command line? Everything I've googled so far has used the command line? BibTeX loses capitals when creating .bbl file Titles of articles I'm about to cite contain upper case letters and when using BibTeX it converts them to lower case ones. This happens only in the title and only the first letter conserves its case. For example, when I cite an article about HF, the reader won't know if it is about Hafnium (Hf) or fluorine acid (HF).  I know that I can fix it manually in the .bbl file but I would like to avoid it, or fix it automatically. I have a Red Hat Enterprise Linux server (7.5 x86_64). I have OpenSSH version 7.4. I was asked to upgrade it to a later version for security reasons: Nessus states that . However the Red Hat software and downloads does not have the latest package RPM.  I found some clues on where to get the latest package for OpenSSH. I found , however, I do not know on how to upgrade it and trust this website. I do not want the SSH and other configuration to be modified by the ugrade.  I did find links but however they are not useful, for example .  I would like to know how to upgrade OpenSSH without using yum.