Determine whether the following series converges: $$\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n^2}.$$   I tried to solve it using ratio's test but it leads me nowhere.
Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent.  I've tried applying all the basic tests to no avail. I need to find out what the "trick" for this particular series is.
Constraint the camera to look at an object In my scene I have a camera sitting at the origin. I would like to contraint it so that it always looks at the center of a given object. But the camera should not change its position.  I found some answers using a "TrackTo" or "LockedTrack" constraint but could not manage to make it work.  Could I probably use a Driver for this and if yes, how can I adjust it?  Thanks in advance! Fractional Laplacian on the torus What is actually the characteristic length in fluid dynamics description? In many fluid-dynamics models and quantities (, ... generally various  descriptions) there is a parameter (usually $L$ or $\ell$) descripted as characteristic linear dimension or characteristic length. What is that in intuitive and physical meaning? I understand the typical examples ( etc.) but I can't see so far any systematic generalisation.  What is the definition? (Or, maybe: What is its role during the derivation?) How do I check if an array includes a value in JavaScript? What is the most concise and efficient way to find out if a JavaScript array contains a value? This is the only way I know to do it: function contains(a, obj) {     for (var i = 0; i &lt; a.length; i++) {         if (a[i] === obj) {             return true;         }     }     return false; }  Is there a better and more concise way to accomplish this? finding rational roots of polynomials Captured variable in a loop in C# Range of A and null space of the transpose of A So I'm complete stuck with something. I know it the following statements are true (or at least the seem to be from the results that I got from messing around with it a bit on MATLAB), but I don't understand why they are true or how to show so. Let $A$ be and $m$X$n$ matrix. Show that:  a) if $x \in N(A^TA)$ then $Ax$ is in both $R(A)$ and $N(A^T)$.   For this one I messed around with it with my own examples and I got $Ax=0$, therefore satisfing the statement, but I don't understand what's actually going on.   b) $N(A^TA)=N(A)$  again, makes sense when I see the results in MATLAB, but don't undestand why it works.  c) $A$ and $A^TA$ have the same rank  d) If $A$ has linearly independent columns, the $A^TA$ is nonsingular.  For the last two I have no idea on how to even start showing the relationship. I feel like I'm missing some crucial relationship between $A$ and $A^TA$, but I'm just not seeing it.   I would greatly appreciate any help of sugestions on how to show that these statements are true.   Thank you very much. What does Pirsig mean by "Quality"? Upgrading from Yosemite to El Capitan, but not Sierra How to get the list of IP addresses in a LAN? C/C++ check if one bit is set in, i.e. int variable int temp = 0x5E; // in binary 0b1011110.   Is there such a way to check if bit 3 in temp is 1 or 0 without bit shifting and masking.  Just want to know if there is some built in function for this, or am I forced to write one myself. Freeze vertical columns in horizonal scrolling visual force page (like excel freeze pane) let's say have 20 columns total.   I want to fix till 5 columns. (means only important ID, etc will be fixed while doing horizonal scrolling)  How can I achieve it in Visual Force Datatable ? Adobe Creative Suite alternative The programs that I am mainly interested in is Adobe Photoshop, Adobe Illustrator (create vector art), Adobe Flash (for animation), Adobe Premiere Pro (edit video), Adobe After Effects (VFX/motion graphics), Adobe Dreamweaver (code HTML/CSS). But can boil down to a video editing software, graphics software and photo editing software.  I understand alternatives such as Inkscape is like Illustrator, Blender is a good substitute for Adobe Flash. Notepad++ for Dreamweaver but if you haven't noticed, these programs are from different companies. What I do appreciate about Adobe is under the master collection, I get all these programs, but I don't appreciate the price (because I can't afford) or the subscription model (I like to buy once and forget) they now have.  I have searched online and can't find something that packs in programs such as Adobe, even if it's just a photo editing software and vector software.  I found Corel Draw, but I'm so confused if they provide a master collection (collection of their software for a price) that will satisfy my needs.  Is there an alternative to Adobe Creative Suite for designers/enthusiast? Interpreting the end of Inception Sorry for the vague title--I don't want the title to be a spoiler.  I've seen the movie three times, and the last two times I looked specifically for clues to this question, but found none.  The final scene in  shows       Cobb's top spinning, as he goes off to be reunited with his children.  The top continues to spin, then the movie ends, without telling us if the top topples--and therefore revealing whether he's in the "real world" or still in a dream. Do we know if he was in a dream at the end of the movie?   I'm assuming the answer is that we simply cannot know.  But did I miss anything?  Were there clues that I missed? What is Kirito's relationship with the girls of SAO? Sending an E-mail to all addresses of all contacts in a contact group in Gmail I'm trying to send an E-mail message using Gmail to a contact group where some of the contacts have more than one E-mail address and I want to send the mail to all of them.   Is there a way for me to do that? I am interested in comparing the computational cost of grid search vs. random search algorithm for hyperparameter tuning. Is random search computationally less expensive then grid search? Is there an article/paper that talks about this topic? Thank you, I'm connecting to some enterprise wireless network (eduroam).  The thing is that it saves the User/Password in the wireless settings, anyone can click 'Show password' and see my university account password.  What I want is that it prompts me everytime the credentials inputs when I try to connect to the network and don't save it under the 'settings' page.  Is this possible? If yes, how?  Thanks  Ubuntu 18.04 LTS Omission of the indefinite article to eliminate ambiguity love between husband and wife   What is the reason that there is no article (e.g. "love between a husband and a wife") in the above? Has some kind of a rule been identified grammatically when it comes to such a case? If I were to guess I think it's because the indefinite article 'a', the purpose of that, is taking a generic sample (or 'instance') from a 'class' (the class of husbands), but in the above sentence you want to refer to the class itself... the husband class and the wife class... and not an 'instance' of them, because it just is more to the point. Do you think this is a fair reasoning? Also, a thought came to me that with the use of the indefinite article, you get a generic husband and a generic wife, which, may not necessarily love each other (for they are not a specific pair of a specific husband and a specific wife who in fact love each other). Prove : $p$(n│even number of ODD parts)=$p$(n│distinct parts ,number of ODD parts is even ) I'm trying to prove the following Integer Partition claim :   $p$(n│even number of ODD parts) = $p$(n│distinct parts ,number of ODD parts is even) .  So I tried to prove a stronger claim :   $p$(n│number of odd parts) = $p$(n│number of distinct parts) :  Using generating functions :   DISTINCT PARTS =  $$(1+x)\cdot(1+x^2 )\cdot(1+x^3 )\cdot(1+x^4 )\cdot\cdot\cdot\cdot$$  = $$\frac {1-x^2} {1-x} \cdot \frac {1-x^4} {1-x^2} \cdot \frac {1-x^6} {1-x^3} \cdot \frac {1-x^8} {1-x^4}\cdot\cdot\cdot\cdot$$  $$ \frac {1} {1-x} \cdot \frac {1} {1-x^3} \cdot \frac {1} {1-x^5} \cdot \frac {1} {1-x^7} \cdot\cdot\cdot\cdot $$  = ODD PARTS  But I'm not sure regarding the proof , does it really prove that even number of odd parts equals distinct parts ,where number of ODD parts is even ?   Thanks