I am wondering if there is a rigourous way to show the following series converges: $$ f(x) = \begin{cases} \dfrac{1}{x^2} &amp; x \in \mathbb{N} \\[1ex] g(x) &amp; x \not\in \mathbb{N}\in \mathbb{R} \end{cases}$$ Here, $g(x)$ could be anything, but for interest, wouldn't converge on its own. When we do the following: $$\lim_{k\to\infty}\sum_{n=1}^{k}f(n).$$ Do we get a limit that exists? Intuitively we would, because summation is only defined on the natural numbers (unless we redefined it here, doing some interpolation between points). However, I am not certain this function is integrable, which would suggest the function does not converge (unless my understanding of the Integral Test is wrong, after all, it says nothing of whether the integral exists, just if it converges the sum converges and vice-versa).
As I have heard people did not trust Euler when he first discovered the formula (solution of the ) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs.   I believe many of you know some nice proofs of this, can you please share it with us?
Is it possible to do a SELECT statement with a predetermined order, ie. selecting IDs 7,2,5,9 and 8 and returning them in that order, based on nothing more than the ID field?  Both these statements return them in the same order:   SELECT id FROM table WHERE id in (7,2,5,9,8)    SELECT id FROM table WHERE id in (8,2,5,9,7) Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$ Can you use Skyrim Legendary Edition mods on the normal edition? I just figured out I had Skyrim in my Steam games. It's the 32-bit non-Legendary Edition Skyrim without any DLCs. Can I still use Legendary Edition mods on it? How to set a domain to ip address? You can reach this IP address  (if my computer is online). I want to assign a domain to it so I wonder how is that possible when I have no DNS? I do own my domain which is ".com". My IP is static.  Thanks The screen of my phone is physically broken, so it doesn't display anything, but the touch aspect does seem to be working, as I can unlock my phone, and get haptic feedback.  Before I go to get it replaced, I'd like to back up some personal data from the built-in memory to the SD card. I'm fairly certain that if I can get adb to recognize the device, then I can use  to see where I am touching the screen and do what I need to do. Problem is, when I plug it in, it's not recognized, while my emulator is.  I suspect that I might not have USB debugging enabled, which I think is necessary for adb to work. If this is the case, can someone give me screenshots of their Droid X (running 2.3.X) going from the home screen, through all the menus, to activating debug mode? Hopefully I can do it by dead reckoning and adb from there.  Edit: I haven't changed anything about the ROM or launcher; it's all stock. Extreme of $\cos A\cos B\cos C$ in a triangle without calculus. If $A,B,C$ are angles of a triangle, find the extreme value of $\cos A\cos B\cos C$.   I have tried using $A+B+C=\pi$, and applying all and any trig formulas, also AM-GM, but nothing  helps.  On this topic we learned also about Cauchy inequality, but I have no experience with it.  The answer according to Mathematica is when $A=B=C=60$.  Any ideas? "It looks straightforward, but actually it isn't" Why can't I transform faces, edges, or vertices without transforming the whole object? I have white blank on the posts.    App Version: 1.0.89 Device Manufacturer: Meizu Device Model: m2   OS Version: 5.1 (1464689330)    I can't see the questions and the answers and the comments.  I saw that it's a common issue..  Does it have a fix?   EDIT -  I saw  answer. This answer is ~2 years old. It says that it will fix in the next update.  His version is 1.0.59 - mine is 1.0.89. Huge different.   The problem is still exists! Find $\sum\limits_{k\, \text{ odd}} \frac{2(k^2-1)}{k^4+k^2+1}$ Closed form solution for slope coefficients in bivariate regression In a univariate regression, $Y=a+bX+e$, the solution for slope b is given by $COV(X,Y)/VAR(X)$.  Is there a similar expression for a bivariate regression $Y=a+bX+cZ+e$. What is the closed form solution for b and c in this case? G is the product of its Sylow subgroups I want to prove the following:     If every Sylow subgroup of $G$ is a normal subgroup, then $G$ is isomorphic to the product of its Sylow subgroups.   So far, I have gotten to realize that for every prime factor $p_i$ of $|G|$, there is exactly one Sylow subgroup and that these form a partition of $G$. However, I am having trouble finding the isomorphism between $G$ and the product which I will call $G_1 \times \ldots \times G_r$. My idea was just to use the mapping $(g_1, \ldots, g_r) \mapsto g_1\ldots g_r$, which is a bijection, but I don't see whether it is a homomorphism. In general, not all elements of $G_i$ commute with elements of $G_j$, so one cannot simply change the order of the elements in the product. What is the usage of Zero Ohm & MiliOhm Resistor? I am new to PCB design and I noticed that some schematics use 0Ω or 100mΩ resistors. What is their purpose and why do we need to use them in our PCB design?  Normally if we wish to probe how much current the load is taking, we put a jumper pin across the PCB trace (then measure current across the pin using a multimeter). Adding resistors for this purpose seems like it would waste a lot of PCB real-estate. Is this the sole reason why 100mΩ resistors are being placed (since I = V/0.1Ω) instead of a jumper pin?   If so, is there any consideration we should take when placing such a mΩ resistor on board so it doesn't affect the signal or behaviour of the circuit? FaceTime not working on iPhone 6 Plus purchased from Dubai I bought my iPhone 6 Plus in Dubai. I am using the phone in Sri Lanka and I cannot use the FaceTime. How can I use FaceTime while in Sri Lanka? Scenario 1: A grimlock hits Alice with a Spiked Bone Club, dealing 5 bludgeoning damage, plus 2 piercing damage. Alice's HP was 12.   Using the optional Massive Damage rules on page 273 of the DMG, does this count as more than half the HP from a single source, or less than half, but twice?  Scenario 2: Assume the same grimlock hits Bob with the same attack, this time rolling 6 bludgeoning damage, plus 3 piercing damage. Bob's Max HP is 8, and he had 1 HP remaining before he was hit.  Using the Instant Death rules on page 197 of the PHB, does the character die, or are the two damage types tracked separately? Does the character have to make a death saving throw for the second part of the damage? Name of short sci-fi story involving a musician who is punished for listening to Bach "Please do not repeat the mistake" and "Please do not repeat the mistake again" Please do not repeat the mistake. Please do not repeat the mistake again.  Should the again be omitted? “Repeat” means “do again”. So to “repeat again” means “to do again again”. Is the second sentence grammatically wrong? When using custom path style in a tikz graph the quotes label isn't printed The $n$ complex $n$th roots of a complex number $z$ What do I need to install LED strip lighting? I've been looking to install LED strips, either RGB or white for lighting and ambiance. It wouldn't be the main lighting of the room, so no necessity for high power LEDs.  The perimeter of the room is a little less than 25 meters. Looking around in online stores, I saw that the reels come in maximum lengths of 5 meters, so I would need five of those.  What I'm not sure is how to connect the strips together.  I was looking at , or  with .  If I choose the white LEDs, what power supply, controller, and connector would you recommend to be able to daisy chain the strips?  If I choose the RGB LEDs, what kind of data repeaters would work to be able to control all the strips at once?