Help with proving this mod statement
Show that conguence $\bmod m\,$ is an equivalence relation
Finding $\tan t$ if $t=\sum\tan^{-1}(1/2t^2)$ I am solving this problem. If $$\sum\limits_{i=1}^{\infty} \tan^{-1}\biggl(\frac{1}{2i^{2}}\biggr)= t$$ then find the value of $\tan{t}$.   My solution is like the following: I can rewrite: \begin{align*} \tan^{-1}\biggl(\frac{1}{2i^{2}}\biggr) &amp; = \tan^{-1}\biggl[\frac{(2i+1) - (2i-1)}{1+(2i+1)\cdot (2i-1)}\biggr] \\\ &amp;= \tan^{-1}(2i+1) - \tan^{-1}(2i-1) \end{align*}  and when I take the summation the only term which remains is $-\tan^{-1}(1)$, from which I get the value of $\tan{t}$ as $-1$. But the answer appears to be $1$. Can anyone help me on this. Permutation count of AABBC Given a string:  $AABBC=A^2B^2C^1$   I am trying to find the Total Permutations (this may be incorrect):  $\dfrac{5!}{2!\cdot2!}=30$  My question is how would I find the partial sums (perhaps the wrong choice in words) when added together would give an index such that:   $AABBC=1$ $ABABC=2$ $CBABA=29$ $CBBAA=30$   (30 might be the wrong number here but is suppose to be the last)   I have tried to draw out a tree structure to get an idea and thought with as the minimum:  $CAABB$  $4(\dfrac{4!}{2!\cdot2!})$ + 1  And:  $CABAB$  I would get something like:  $4(\dfrac{4!}{2!\cdot2!}) + 0 + 1(\dfrac{2!}{2!}) + 0 + 0$  But I am missing something here. How would I find the sums needed for $BACAB$ ?  More specifically I am trying to find the index of a word that is arbitrarily long in alphanumeric.  So the string may also be something like:  $AAAAABBBBDDEEFFFGGGGGHIIIIJJJJKLMOOOPZZZZZ$ New Div - "Hiding questions because of your Ignored Tags preferences for tags" I think this message/div/box was just added today since I've never noticed it before, it is very noticeable.    I'm already aware that I'm hiding questions because I selected to hide them in my preferences. Isn't this a bit redundant? Now I have to scroll down more to be able to click to the next page. Plus see that same message dozens upon dozens of times a day.    Here are my ignored tags. So I'm going to have to see the same message on any question page that I'm on. Even though I already know questions are being hidden. I'm not understanding the point to this new div/box notification.  Does anyone find this new div/box useful? Do you find it redundant?       EDIT:   Also, this isn't just for StackOverflow.   I only have 2 ignored tags for CodeReview currently. And I have it set for only 15 questions per page. But still, on every page I click to it shows! In preparation for the eventual deployment of full HTTPS support on Stack Exchange, and also to make the currently available "alpha" HTTPS support more usable, all links from Stack Exchange sites to other Stack Exchange sites should be .  Currently, at least the following places still feature links with explicit http: protocol prefixes:   the page footer, the "Hot Network Questions" and "Community Bulletin" modules in the sidebar,  (the site switcher menu is OK, though),   and, of course,   lots and lots of user-supplied links in post and comment content.   I've posted this feature request with the intent to provide a single central place for tracking these issues, rather than having them scattered across multiple meta threads.  If you find more places on SE where protocol-relative links should be used, please report them below.  I fully expect this request, in its full sense, to be treated as  for the time being; however, these issues do need to be fixed at some point before our HTTPS support is ready for "prime time".  Ps. Yes, the next release of  (v1.10) is going to include a general client-side fix for most of these issues.  In the mean time, the  plugin effectively provides the same functionality, and much more besides. Known proofs of Wirtinger's Inequality? How to calculate stays against 90/180 visa rules? Some destinations have a "90/180 rule" which says you can come and go but you must not spend over 90 days out of any 180 days in the area.  concerns this rule.  At least Schengen and Turkey have this rule, probably depending on where the traveller is from and what kind of visa they apply for.  Maybe I'm a dummy but I've always had a lot of trouble trying to add up how many days I've spent in such places when I've entered and left several times over recent months.  Is there an easy way to do this I'm missing or is it difficult for everybody? I'm guessing something based on counting backwards from the end date might be the trick?  Maybe you know of an app or website that can do this for you. I recently upgraded to Ubuntu 20.04 lts from 18.04 lts. After reboot audio is not working. I tried to reinstall pulseaudio and alsa but still not working. sudo apt-get install --reinstall alsa-base pulseaudio sudo alsa force-reload still not working. output  └─▶ inxi -SMA System:    Host: Wizard Kernel: 5.4.0-26-generic x86_64 bits: 64 Desktop: Gnome 3.36.1 Distro: Ubuntu 20.04 LTS (Focal Fossa)  Machine:   Type: Portable System: Dell product: Vostro 2520 v: N/A serial: &lt;superuser/root required&gt;             Mobo: Dell model: 0V1NDP v: A11 serial: &lt;superuser/root required&gt; BIOS: Dell v: A11 date: 02/24/2014  Audio:     Device-1: Intel 7 Series/C216 Family High Definition Audio driver: snd_hda_intel             Sound Server: ALSA v: k5.4.0-26-generic     output └─▶ lspci  00:00.0 Host bridge: Intel Corporation 3rd Gen Core processor DRAM Controller (rev 09) 00:02.0 VGA compatible controller: Intel Corporation 3rd Gen Core processor Graphics Controller (rev 09) 00:16.0 Communication controller: Intel Corporation 7 Series/C216 Chipset Family MEI Controller #1 (rev 04) 00:1a.0 USB controller: Intel Corporation 7 Series/C216 Chipset Family USB Enhanced Host Controller #2 (rev 04) 00:1b.0 Audio device: Intel Corporation 7 Series/C216 Chipset Family High Definition Audio Controller (rev 04) 00:1c.0 PCI bridge: Intel Corporation 7 Series/C216 Chipset Family PCI Express Root Port 1 (rev c4) 00:1c.3 PCI bridge: Intel Corporation 7 Series/C216 Chipset Family PCI Express Root Port 4 (rev c4) 00:1c.5 PCI bridge: Intel Corporation 7 Series/C210 Series Chipset Family PCI Express Root Port 6 (rev c4) 00:1d.0 USB controller: Intel Corporation 7 Series/C216 Chipset Family USB Enhanced Host Controller #1 (rev 04) 00:1f.0 ISA bridge: Intel Corporation HM75 Express Chipset LPC Controller (rev 04) 00:1f.2 SATA controller: Intel Corporation 7 Series Chipset Family 6-port SATA Controller [AHCI mode] (rev 04) 00:1f.3 SMBus: Intel Corporation 7 Series/C216 Chipset Family SMBus Controller (rev 04) 07:00.0 Network controller: Broadcom Inc. and subsidiaries BCM43142 802.11b/g/n (rev 01) 09:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 07) How to select all faces with the same material? Search all tags with no tag wiki Is there a way I can search the Tags for tags with no tag wiki created yet? Opacity problem over white object in Illustrator I have a problem. I'm working with an object at 50% opacity. When it's over the artboard itself the color looks great, but when I put a white object behind it I get this weird color that doesn't look accurate.  This happens with every object and only in CMYK. I'm pretty sure this never happened to me before so maybe I changed something in the settings or I don't know.  Does anyone have any idea? How can I check dependency list for a deb package pronoun agreement in neither nor Which one is correct?   Neither my friend nor I feel my best in the morning. Neither my friend nor I feel her best in the morning. Neither my friend nor I feel our best in the morning.    I already know that the verb agrees with the subject closer to the verb:   Neither my friends nor my brother understands me. (correct) Neither my friends nor my brother understand me. (wrong)   AND   Neither my brother nor my friends understand me. (correct) Neither my brother nor my friends understands me. (wrong) can CakePHP automatically create tables from models? Will wolves attack and kill ocelots in Minecraft? I know they kill sheep. I just don't want my cats being eaten. I have the following ammeter:   The seller recommends 75mV 100A shunt be placed with it. I want to know what will happen if I change the voltage going through the shunt. Should I be changing the shunt as well to read amps correctly?   I want to build a 1-40V (or maybe 50V) adjustable DC power supply and use this ammeter to measure current. What the right value of shunt that I should use to get the right measurement? Changing shunts in a lab bench power supply is impractical, of course.  EDIT: I am planning to measure only up to 5A or so. Sorry I forgot to mention this previously. How to solder with one hand? I am a beginner at electronics.  I don't have many tools and stuff, so I am forced to solder by hand.   I don't have a pcb to solder on, so I usually just solder the wires together, but it's really difficult without a holder.  Is there an easier way to do this? If not, is there a substitute for a wire holder?  Any help would be appreciated. Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $? Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?   I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this.  Thank you. Chrome asks for password to unlock keyring on startup In Google Chrome, when I go to a login page, a window pops up asking to "Enter password for keyring 'default' to unlock". In most cases, whether I click Cancel or enter my password, the login form gets auto filled anyway.  How do I get rid of the popup? I want it to auto login each time, not ask for my system password. The dialog box never appears for any other apps. I found this , but I am not sure if this is the same issue.  On , I did some reformatting of the indentation in the code block to make it a bit more readable and it was  for peer review. A few minutes later, it appears the OP made an edit that added some additional code. It looks like my suggested edit was applied when the OP saved his/her edit, and it looks like he got the "credit" for the edit.  I'm not really concerned about the 2 rep points; I'm more curious why this happened and if this is a bug (or limitation) in the editing feature, or if this is known/intended behavior?  Note that this is not a suggested edit that's overridden by a concurrent edit: the recorded edit starts from the post-suggested edit state (and in fact can be identical to it, as in  ). Windows 7 / 8 stop modal dialogue flash on parent focus Is there some setting in Windows where I can change the default behavior that causes the border and title of a modal dialogue box to flash like 6 or 7 times when I click on the parent window?  Simply focusing it would be plenty to let me know what I did was wrong.   I don't feel the need for all the sirens and bull horns and strobe lights to let me know I clicked in the wrong place.