Could someone please help me with how do I calculate the sum of the $$\sum_{n=1}^{\infty}\frac{1}{4n^{2}-1}$$ infinite series? I see that $$\lim_{n\rightarrow\infty}\frac{1}{4n^{2}-1}=0$$ so the series is convergent based on the Cauchy's convergence test. But how do I calculate the sum? Thank you.
I am trying to calculate the sum of this infinite series after having read the series chapter of my textbook: $$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}$$  my steps:   $$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}=\sum_{n=1}^{\infty}\frac{2}{4n^2-1}-\sum_{n=1}^{\infty}\frac{1}{4n^2-1}=..help..=sum$$  I am lacking some important properties, I feel I am coming to the right step and cannot spit that out..
Is Canon 60D still a decent camera? I have a Canon 60D, bought it back in 2013. Use it quite a while, but in past two-three years, I lost interest in photography. I will be going on a long trip this summer and would like to take a decent camera with me. Is Canon 60D still a decent one or I will be missing on features and upgrades comparing to the latest versions?  In terms of lenses I own Tamron 24-70 2.8 And Canon 100 2.8 Macro. How can I obtain a Cauchy distribution from two standard normal distributions? I am interested in      Let $X\sim N(0,1), Y \sim N(0,1)$ independently. Show $\frac{X}{X+Y}$   is a Cauchy random variable.   My work:  $f_{X,Y}(x,y)=\frac{1}{2\pi} e^{\frac{-1}{2}(x^2+y^2)}, -\infty&lt;x,y&lt;\infty$ by independence  Let $U=\frac{X}{X+Y},V=X+Y$. (Is there a better $V$ to choose for this bivariate transformation?)  Then, $X=UV, Y=V - UV$. So, $|J|=V$.  $f_{U,V}(u,v)=f_{X,Y}(uv,v-uv)|J|=\frac{v}{2\pi}e^{\frac{-v^2}{2}(2u^2-2u+1)},-\infty&lt;u&lt;\infty$  $f_U(u)=\frac{1}{2\pi}\int_{-\infty}^{\infty}ve^{\frac{-v^2}{2}(2u^2-2u+1)}dv$. Let $y=\frac{v^2}{2}(2u^2-2u+1)$, so $dy=v(1+2u^2-2u)dv$. Then,  $f_U(u)=\frac{1}{2\pi(2u^2-2u+1)}\int_0^{\infty}e^{-y}dy=(\pi[(\frac{u-1/2}{2})^2+1])^{-1}, -\infty&lt;u&lt;\infty$,   which is not exactly a Cauchy distribution. Where did I mess up? More importantly, how would you proceed in solving this problem? The space obtained by identifying the antipodal points of a circle Can we have a moderator only "flag for community team/developer attention" link on posts? There are times when moderators come across things that only Stack Exchange employees can deal with. These include things like deleting sensitive information permanently from posts disassociating users from posts that require direct database access.  (The previous use case is now possible for moderators to do directly the moderator redaction tool)  Currently the workflow is:   See the post (either in normal site usage or via a flag) Go to chat Ping one of the community team (if they are around) Explain the problem   A far better workflow for the moderator would be to simply able to flag the post so it then appeared in an "employee only" queue somewhere. This could either be a page on the main Stack Exchange site or (better still) their inbox.  (OK I admit I hadn't fully thought this last bit out.)  This could either be a moderator "flag to devs" link on the post or a new option on the existing flag dialog (probably the better solution).  Now that moderators can't merge we have a fairly common situation where this would be very useful. A new users posts an answer (or question on sites that don't require registration) then loses their cookies and ends up creating another unregistered account (or two). Eventually the might even create a registered account. It used to be easy to merge the unregistered accounts into the registered ones, but now we either have to badger the OP to use the "merge accounts" form or ping the team somehow.  Having the "flag for team" option would be very useful in this scenario. A new Victorian-set British novel begins sentences containing h-dropping with lower-case e's, as in " 'e took my money, 'e did." This seems incorrect but I'm at a loss to find the rule. mbstring is missing for phpmyadmin in ubuntu 16.04 When I installed phpmyadmin in Ubuntu 16.04 LTS, it gives me an error mentioned below.     The mbstring extension is missing. Please check your PHP configuration   then I tried to install php-mbstring, it says mbstring is up to date, Can anyone help me to solve this? How to deserialize a JSON String to Apex How can I deserialize this json object:      {     "response": {         "count": 1,         "benchmark": 0.22567009925842,         "requests": [             {                 "request": {                     "id": 537481,                     "image_thumbnail": "",                     "title": "Request for new bin(s) - residential",                     "description": "Propmain ref  3234-1114",                     "status": "submitted",                     "address": "36 Pine Tree Close",                     "location": "Peterborough, England",                     "zipcode": "PE1 1EJ",                     "user": "",                     "date_created": 1417173208,                     "count_comments": 0,                     "count_followers": 0,                     "count_supporters": 0,                     "lat": 52.599967,                     "lon": -0.233482,                     "user_follows": 0,                     "user_comments": 0,                     "user_request": 1,                     "rank": "0"                 }             }         ],         "status": {             "type": "success",             "message": "Success",             "code": 200,             "code_message": "Ok"         }     } }   What i've tried:   Map&lt;String,Object&gt; rawObj = (Map&lt;String,Object&gt;) JSON.deserializeUntyped(jsonString);     Map&lt;String,Object&gt; responseObj = (Map&lt;String,Object&gt;)rawObj.get('response');     List&lt;Object&gt; reqs = (List&lt;Object&gt;) responseObj.get('requests');     System.debug('Map Size = ' + reqs.size());     Map&lt;String, Object&gt; i = new Map&lt;String, Object&gt; ();     for (Object x : reqs) {          i = (Map&lt;String, Object&gt;)x; }     Map&lt;String,Object&gt; requests = (Map&lt;String,Object&gt;)i.get('request');     System.debug('Map Size = ' + i.size());     for (String field : i.keySet()){          Object id = i.get(field);         Object title = i.get('title');         System.debug('Id : ' + id);         System.debug('title : ' + title);          //System.debug('Title : ' + title);       } Basically, I have a kindle fire that uses a form of the android OS and was wondering if there was a way I could run similar apps on my Ubuntu system. Specifically I'd like the news reader program "Pulse".  Anybody know if such an emulator or workaround exists? I installed blender for the first time, so I have the latest version, 2.78c.  I played with it for a bit.  Then I discovered I couldn't save my work.  I uninstalled it and reinstalled it.  Rebooted.  Same problem.  After a little testing, I realized I CAN save to my NAS storage drive.  I cannot save on my C: drive.  Any project.  Any folder.  C: always fails.  If I run blender as Administrator, it will save to the C: drive.  But not as my normal userid.  So it sounds like a bug of some kind.  I am running Windows 10.  Bitdefender A/V.  I use OneDrive.  I have a 2TB drive installed.  All clean and purring nicely.  Is there a better way to install it?  Or some change I can make (other than always running as administrator)?    Sounds like a bug - but hopefully there's a proper solution.  Thanks,  Jerry Very specific paragraph shape Preventing Brute Force Logins on Websites As a response to the recent  and , what is the best way to secure your website against brute force login attacks?  Jeff's post suggests putting in an increasing delay for each attempted login, and a suggestion in the comments is to add a captcha after the 2nd failed attempt.  Both these seem like good ideas, but how do you know what "attempt number" it is?  You can't rely on a session ID (because an attacker could change it each time) or an IP address (better, but vulnerable to botnets).  Simply logging it against the username could, using the delay method, lock out a legitimate user (or at least make the login process very slow for them).  Thoughts? Suggestions? Using global variables in a function Sudo not available in guest login Possible Duplicate:          The  describes the expansion of the $n$th power of the sum $(a+b)$:  $$(a+b)^n = \sum_{k = 0}^n {n\choose k}a^kb^{n-k}$$  In calculus, there is a generalization of the product rule called , which describes the expansion of the $n$th derivative of the product of two functions $f$ and $g$:  $$(fg)^{(n)} = \sum_{k = 0}^n {n\choose k}f^{(k)}g^{(n-k)}$$  It's not difficult to prove using the product rule and induction, what I'm really trying understand here is why the formula is true. Is there a way to prove it using the binomial formula? How are these two formulas related? Is there a combinatorial proof for Leibniz's rule like there is for the ?   I understand this question is a bit vague, but I hope you'll agree that these two formulas look so similar it just feels like there should be some relation between the two. Of course this is not necessarily true, but I'm hoping it's not just a crazy coincidence. How to setup OpenVPN Client A real nxn matrix always has a real eigenvalue when n is odd what do it mean by "A real nxn matrix always has a real eigenvalue when n is odd?" Shouldnt it be that regardless of odd or even there will always have a real root?  hope to hear from you guys soon:) I know this may be a tough or confusing question, but I have numerous existing materials I created and reuse for a project, which were all prepared for Blender Internal.  Now, I want to start the transition to Cycles, but I've hit a roadblock. How can I transition my materials without completely starting over? Or should I just forget it and start with fresh materials? Splitting all polygon features in a shapefile into smaller equal width polygons I have a shapefile of thousands of polygon features, that i would like to break up into smaller pieces (i.e. break a square into 8 equal pieces). I have read all over these boards for the right solution and haven't found one yet unfortunately. I've tried parcel editor but that only allows you to manually edit one feature at a time, and I've tried fishnet but that doesn't run through each individual feature it just processes across the entire extent of the shapefile. I have the full license for ArcGIS so all options are game. Ultimately I would like to be able to split this initial shapefile of features into a specified amount of equal width slots that each would be their own feature in the resulting shapefile.  Any thoughts? So I want to texture clothing for example that I created using the mirror modifier. But I want the 2 sides of the clothing piece to have individual designs for instance but it is not possible since the original and the mirrored surfaces use the same UV-coordinates. How can I change that? Batch/Multi rename only files with a specific number range I have a situation where I have the following:  file01.ext . . . file99.ext file100.ext . . .   And I want to (using rename ) insert the '0' for the files 01-99 (so that they become 000-099) but not modify files larger than 100. Which expression should I use and what is a good source on learning regular expressions such as these?   On Windows I would have used Total Commander's multi rename function which has a very simple renaming syntax. Are there some good alternatives for that tool (multi-rename) for Ubuntu?