So, the question is as follows: Prove that $\left|\sin 1\right| + \left|\sin 2\right| + \left|\sin 3\right| +\cdots+ \left|\sin 3n\right| &gt; 8n/5.$ I have tried quite a few approaches, including using the Taylor expansion or the A.M-G.M inequality, but to no avail. It would be of great help if this problem could be proven. Edit: Angles are in radians, $n$ is a natural number.
I want to show for $x\in\mathbb{R}$, the following inequality holds. $$ \left|\sin x\right|+\left|\sin (x+1)\right|+\left|\sin (x+2)\right|&gt;8/5. $$ This trigonometric inequality has been verified by Mathematica using the Plot commend. However, using the sum formula for the sine function seems doesn't work, and I cannot give a rigorous proof of it. Any suggestion, idea, or comment is welcome, thanks!
How to find out the reason, why my PC can't run Windows 11? Windows 11 requires this features:  Processor 1 GHz or faster with 2 or more cores on a compatible 64-bit processor or System on a Chip (SoC) Memory 4 GB RAM Storage 64 GB or larger storage device System firmware UEFI, Secure Boot capable TPM Trusted Platform Module (TPM) version 2.0 Graphics card DirectX 12 compatible graphics / WDDM 2.x Display &gt;9” with HD Resolution (720p) Internet connection Microsoft account and internet connectivity required for setup for Windows 11 Home  I have a 15-ay108nh laptop. As far as I know, it supports all the required features. It has 8GB ram, 256GB storage, UEFI support, TPM 2.0 ( I enabled it in the BIOS ), DirectX 12 compatible graphics / WDDM 2.x, and 1080p screen. I installed the , but for some reason, I get this result:  How is it possible to find out exactly, why it fails? Can Reduce *really* not solve for x here? Splitting raster into smaller chunks using GDAL? I have a raster (USGS DEM actually) and I need to split it up into smaller chunks like the image below shows. That was accomplished in ArcGIS 10.0 using the Split Raster tool. I would like a FOSS method to do this. I've looked at GDAL, thinking surely it would do it (somehow with gdal_translate), but can't find anything. Ultimately, I'd like to be able to take the raster and say how large (4KM by 4KM chunks) I would like it split up into. Left and right subscript / superscript I am trying to put two subscripts at the left and right of a character. For example, something like:  _{t} p_{x}  where p is in the middle. How do you do this? Getting polygon shapefile node coordinates and point order I have been denied a visa several times in a row. How does one deal with such serial refusals? Over the course of time here on TSE, I have observed individuals whose pattern of visa applications is best described as ''...     serial refusals - Two or more consecutive visa refusals issued by the same regime to the same person. Commonly for the same, or nearly the same, reasons. In some visa regimes Serial refusals can have a cumulative effect wherein the applicant's credibility is increasingly eroded, especially when the application pattern suggests a secondary agenda.   A common thread appears to be the need to visit the destination country (especially Schengen, the USA, or the UK) with a special sense of urgency. Also common in these cases are the following types of questions...   Should I just reapply once again? What are my chances? What can I do to improve my odds in the future? Can I somehow address the previous refusals to avoid them being a burden on all future applications? How can I break the pattern?      NB: This question is intended as a canonical about serial refusals. I have never been personally denied a visa. Suppress "In:" biblatex How do I suppress the "In: " before the journaltitle? Like so:     In: Journal of Applied Physics I need rules to drop some malicious Apache connections Someone asked me this question and I don't think I gave him an adequate answer (I was trying to think of the extreme case - that of neutron stars) Glucagon stimulates glycogenolysis and gluconeogenesis, thus increasing the plasma glucose concentration — so that tissues get enough glucose in the fasting state. However glucagon also inhibits glycolysis. In this case, what is the point of having high blood glucose if it cannot be catabolized? Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition Let $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ be a subset of $\mathbb{R}$. I have to prove using the open cover definition that this is compact. Could you help me, please? If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric Change style of labels of Tikz nodes of a given style Why isn't "Column ordering" available in my list settings? I want to change the column order visible when viewing or editing a list item. I have accessed the "Column ordering" interface via the list settings page on other lists in the past, but it isn't visible in this list's settings. Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$. I made a simple FM transmitter following  and it worked! Now I need some help to understand how.  My coil is made of 1 mm copper wire has 6 turns and 8 mm wire diameter. According to this online calculator:  this means that: L = 3.9e-7 H  I did not have any 5 pF trim capacitor so I used a 3 pF capacitor instead.  Am I correct in assuming that the frequency this will transmit on can be found by looking at the L1 C2 as a LC oscillator and using the frequency formulae:  fR = 1/(2*pi*sqrt(LC))  ?  If so I should get fR = 147 MHz. Which is strange since I received the signal on 91.5 MHz on my radio.  In a LC circuit I understand that the signal decays exponentially over time. That is hopefully not happening here. Why? Because of C4 maybe?  Also what is the purpose of C1?  If I want to increase the range of this FM transmitter somewhat can I do this by simply replacing R2 with a 470 Ohm resistor?  Are there free simulator programs that let me see the resulting spectrum at the antenna? What's the best practice of inserting a collection of data into a SQL Server database Remove pseudo nodes in QGIS Bounds for $n$-th prime In  I have found that the bounds for $n$-th prime is given by, $$n(\ln n+\ln \ln n)&gt;p_n&gt;n(\ln n+\ln \ln n-1)$$ for all $n\ge6$. Are there even stronger bounds for the $n$-th prime?   If possible (of course if the answer is affirmative) in the answer (or comment) please give the link of the paper in which it first appears. Hot questions no longer work on iPad (ok on iPhone)