How we can do this sum? $$ \sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90} $$  I know that we could possibly use a Fourier series decomposition however I don't know what function to start with.  I know it is also equal to $\zeta(4)$ but a proof involving this is tough for me since they don't teach us these at school.  Thanks.  Thanks
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$.  For example, see  or the answers to the question "?"  Are there any nice ways to prove that $$\zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}?$$  I already know some proofs that give all values of $\zeta(n)$ for positive even integers $n$ (like #7 on Robin Chapman's list or Qiaochu Yuan's answer in the linked question).  I'm not so much interested in those kinds of proofs as I am those that are specifically for $\zeta(4)$.  I would be particularly interested in a proof that isn't an adaption of one that $\zeta(2) = \pi^2/6$.
Is Iron Man stronger than the Hulk? In Avengers: Endgame, when Hulk initially uses the new Infinity Gauntlet, it takes a tremendous effort from him (in addition to almost burning his entire arm), in order to snap his fingers and restore the world as we knew it. Yet at the end of the movie, when Iron Man uses the Gauntlet, it seems very easy for him to snap his fingers.  How was Tony able to wield the stones with such ease when Hulk was clearly struggling with it? What's a concise way to check that environment variables are set in a Unix shell script? An integer is a $n$'th power if that holds true for all moduli. I have not been able to solve this problem. Any insights would be appreciated!   Let $x, n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_{k}$ such that $x − a_k^n$ is divisible by $k$. Prove that $x = A^{n}$ for some integer $A$. Smoothing/interpolating across images in an ImageCollection to remove missing data I have some code that is working properly to calculate the monthly NDVI averages for the previous two years from a date but I sometimes have missing data in some of the months and I would like to get an average based on the mean of the previous two months. I suppose I could also interpolate from the closest geographic data point but it seems like it would be better to interpolate across time than distance given that sometimes the clouds cover reasonably large areas.   But I'm certainly open to other opinions, I just want to fill these holes as accurately as possible. Is there an efficient way to do this.  and here is the code itself:  // pick a landsat tile footprint to use as my geometry var wrs2_descending = ee.FeatureCollection('ft:1_RZgjlcqixp-L9hyS6NYGqLaKOlnhSC35AB5M5Ll'); // use a manually defined point to pick the WRS2 tile var wrs2_filtered = wrs2_descending.filterBounds(roi); var layer1 = ui.Map.Layer(wrs2_filtered, {}, 'WRS2 filtered'); Map.layers().set(1, layer1);  var imageCollection = ee.ImageCollection('LANDSAT/LC08/C01/T1_SR').filterBounds(wrs2_filtered); var monthCount = ee.List.sequence(0, 11);  // Function to cloud mask from the pixel_qa band of Landsat 8 SR data. function maskL8sr(image) {   // Bits 3 and 5 are cloud shadow and cloud, respectively.   var cloudShadowBitMask = 1 &lt;&lt; 3;   var cloudsBitMask = 1 &lt;&lt; 5;    // Get the pixel QA band.   var qa = image.select('pixel_qa');    // Both flags should be set to zero, indicating clear conditions.   var mask = qa.bitwiseAnd(cloudShadowBitMask).eq(0)       .and(qa.bitwiseAnd(cloudsBitMask).eq(0));    // Return the masked image, scaled to reflectance, without the QA bands.   return image.updateMask(mask).divide(10000)       .select("B[0-9]*")       .copyProperties(image, ["system:time_start"]); }  // run through the image collection and generate monthly NDVI median images var composites = ee.ImageCollection.fromImages(monthCount.map(function(m) {   var startMonth = 1; // thinking that I should always start from Jan so the series are similar   var startYear = ee.Number(2017-1); // number is one year before the current one    var month = ee.Date.fromYMD(startYear, startMonth, 1).advance(m,'month').get('month');   var year = ee.Date.fromYMD(startYear, startMonth, 1).advance(m,'month').get('year')    // filter by year and then filter by month to get monthly mosaics   var filtered = imageCollection.filter(ee.Filter.calendarRange({     start: year.subtract(1), // we want an average of the last two years     end: year,     field: 'year'   })).filter(ee.Filter.calendarRange({     start: month,     field: 'month'   }));   // mask for clouds and then take the median   var composite = filtered.map(maskL8sr).median();   return composite.normalizedDifference(['B5', 'B4']).rename('NDVI')       .set('month', ee.Date.fromYMD(startYear, startMonth, 1).advance(m,'month')); })); print(composites);  Map.addLayer(composites, {min: 0, max: 1}, 'check');  // stack the ImageCollection into a multi-band raster for downloading var stackCollection = function(collection) {   // Create an initial image.   var first = ee.Image(collection.first()).select([]);    // Write a function that appends a band to an image.   var appendBands = function(image, previous) {     return ee.Image(previous).addBands(image);   };   return ee.Image(collection.iterate(appendBands, first)); }; var stacked = stackCollection(composites); print('stacked image', stacked);  // Display the first band of the stacked image. Map.addLayer(stacked.select(0).clip(wrs2_filtered), {min:0, max:1}, 'stacked'); Help proving the primitive roots of unity are dense in the unit circle. I'm having difficulty understanding how to prove that the primitive roots of unity are in fact dense on the unit circle. I have the following so far:  The unit circle can be written $D=\{x\in\mathbb{C}:|x|=1\}$. The set of primitive $m$-th roots of unity is $A_m=\{\zeta_k:\zeta_k^m=1,\zeta_k\text{ is primitive}\}$.  Hence, the set of all primitive roots $A$ is given by the union of $A_m$ over $m=1,2,3,\ldots$. But I can't seem to get started on how to prove that $A$ is dense in $D$. Why is my USB drive showing corrupted data when plugged as an internal SATA drive? I have a 3TB Seagate Backup Plus Desktop USB 3.0 drive, which works fine when in its enclosure, but when I get it off its enclosure and directly plug it as an internal SATA drive, it's just not properly recognized (it works again when used in the enclosure). My systems’s motherboard is an , which has two SATA 6.0 Gb/s and four SATA 3.0 Gb/s ports.  When used as an internal SATA drive Windows 7 asks if I want to format the drive, as if it didn’t have a proper filesystem, and if I use the DISKMGMT.MSC tool (typing that in the start menu) I get completely wrong information about the drive.  It says the drive has 3 partitions (349,31GB of RAW data and two unassigned partitions of 1698,68GB and 746,52GB). This information is plain wrong, since the drive, when used in the USB enclosure, works as a single NTFS partition (2794,52GB). How come it’s shown as 3 partitions without filesystem when connected through SATA?!  Is Seagate using some proprietary way of storing the data when using the drive in its provided USB 3.0 enclosure? I didn’t use any kind of encryption. I just plugged the drive with USB 3.0 and started using it straight away, since it apparently came pre-formatted with an NTFS filesystem. I need to cast single figures (1 to 9) to (01 to 09). I can think of a way but its big and ugly and cumbersome. I'm sure there must be some concise way. Any Suggestions Cropping Shapefiles in QGIS? I recently got a lenovo Y40 laptop and after i installed ubuntu 14.04 (also in the live usb) Wireless doesn't work. It says that it is disabled by hardware switch, however the hardware switch doesnt work (its intention is to put on airplane mode but does nothing when pressed in ubuntu) wireless works in windows and the BIOS has it enabled. I tried resetting the BIOS but it still doesn't work. Here are some command outputs  lspci:  03:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 10) 04:00.0 Network controller: Intel Corporation Wireless 3160 (rev 93)   rfkill list all:   0: ideapad_wlan: Wireless LAN     Soft blocked: no     Hard blocked: yes 1: ideapad_bluetooth: Bluetooth     Soft blocked: no     Hard blocked: yes 2: hci0: Bluetooth     Soft blocked: no     Hard blocked: no 3: phy0: Wireless LAN     Soft blocked: no     Hard blocked: no   lswh:     *-network         description: Ethernet interface         product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller         vendor: Realtek Semiconductor Co., Ltd.         physical id: 0         bus info: pci@0000:03:00.0         logical name: eth0         version: 10         serial: f8:a9:63:31:2e:b6         size: 1Gbit/s         capacity: 1Gbit/s         width: 64 bits         clock: 33MHz         capabilities: bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt 1000bt-fd autonegotiation         configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=full firmware=rtl8168g-3_0.0.1 04/23/13 ip=192.168.1.4 latency=0 link=yes multicast=yes port=MII speed=1Gbit/s         resources: irq:64 ioport:4000(size=256) memory:c0604000-c0604fff memory:c0600000-c0603fff *-pci:3      description: PCI bridge      product: Lynx Point-LP PCI Express Root Port 4      vendor: Intel Corporation      physical id: 1c.3      bus info: pci@0000:00:1c.3      version: e4      width: 32 bits      clock: 33MHz      capabilities: pci normal_decode bus_master cap_list      configuration: driver=pcieport      resources: irq:59 memory:c0500000-c05fffff    *-network DISABLED         description: Wireless interface         product: Wireless 3160         vendor: Intel Corporation         physical id: 0         bus info: pci@0000:04:00.0         logical name: wlan0         version: 93         serial: a0:88:69:90:0a:38         width: 64 bits         clock: 33MHz         capabilities: bus_master cap_list ethernet physical wireless         configuration: broadcast=yes driver=iwlwifi driverversion=3.13.0-32-generic firmware=22.24.8.0 latency=0 link=no multicast=yes wireless=IEEE 802.11abgn         resources: irq:66 memory:c0500000-c0501fff I read this at least 20 years ago, and I believe it was originally published in the '60s.   The main character is a human, but somehow "inherits godhood" from an alien, whose species is very long lived. IIRC this is one of the old-style god thingies, i.e. "god of thunder" not "omniscient omnipotent omnipresent transcendent being".  The plot somehow revolved around identifying and fighting some enemy which, I believe, was in turn a member of the alien species who acted as the avatar of the opposite god in the alien pantheon.  The only thing I remember for sure is that the enemy was identified as "green green" or "gringrin" (sadly, I read this in translation so I am not sure what the original meant) Personal Hotspot not working over Wi-Fi on iOS 13 After upgrade to iOS 13 from 12.4, I noticed that personal hotspot isn't working anymore over Wi-Fi (via USB is ok).  Tested on two computers with Win7. The PC can connect to Wi-Fi, but no valid IP (IPv4) is assigned and no connection is working to internet. Do Triggered effects stack? Two-Weapon Fighting & Bonus Actions I've always been confused by the differences in behavior of font commands.  For example, we have \mathrm{} and \texttt{}, which can be used like this: \texttt{hello} world.  And then we have \it and \bf which must be used as {\it hello} world. If one uses \it{hello} world, both words are italicized.  Internally, what is the difference here? And was this difference a conscious design decision? If so, why? Objects vs arrays in Javascript for key/value pairs What is the resolution of tapping a blocking creature? Let's say I have declared a blocker during the correct phase and during the activate abilities step I choose to tap the blocker to use its ability.  All the abilities resolve and now we move to the declare damage step.  I believe the resolution is that the attacking creature remains blocked and that it deals damage to my blocker; however, the blocker does not deal damage to the attacker.  How would this change if the blocker had first strike? How would this change if the blocker was sacrificed? Inverse function of $x^x$ How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried using logs, differentiating, etc, etc, but to no avail. Why can't you use the keyword 'this' in a static method in .Net? \bar vs. \overline - when to use what, semantically? Following the question "" which is simply about the lengths of both, I'd like to know when to use \bar and when to use \overline from a semantically viewpoint.  I always used \bar, because up until now I only had single letter mean values to type. But now I came to a problem where I needed a wider "bar".  Then I just tried  \documentclass{article}  \begin{document}   \[ \bar{a} \overline{a} \]   \[ \bar{abc} \overline{abc} \] \end{document}   where one can see that they also differ in position (\overline is lower), and also in thickness (\overline is thicker).  So, the question here is:  What does \bar represent semantically? And when to choose it over \overline and vice versa? What does the register keyword do in C language?  I have read that it is used for optimizing but is not clearly defined in any standard.  Is it still relevant and if so, when would you use it?