Since your ultimate goal is to make a chart, you may want to pivot the data so that each fruit is in its own column.

Assuming the data is in Sheet1!A2:B, choose Insert > New sheet and enter this array formula in cell A2 of the new sheet:

=query(
Sheet1!A2:C,
"select A, max(C)
where A is not null
group by A
pivot B",
0
)

Then create a chart based on the data in the new sheet.

If you really need to expand the data so that every date and fruit has a value, and that value is zero when no value appears in the source data, insert yet another sheet and enter this array formula in cell A2 of the new sheet:

=arrayformula( query( split( flatten( unique(trim(Sheet1!A2:A)) & "µ" & transpose(unique(Sheet1!B2:B)) ), "µ" ), "where Col1 is not null and Col2 is not null", 0 ) )

Let $0<p<1$ be a positive real number strictly smaller than one and $q>0$ be a positive real number. Consider the series $$
mathsf{Li}_{-q}(p) = sum_{ell=1}^{+infty}ell^{q}p^{ell}
$$
which defines the polylogarithmic function. Is there any result on the rate at which the reminder $$
c_n=sum_{ell=n}^{+infty}ell^{q}p^{ell}
$$
goes to zero? My guess is $c_n=O(n^q,p^n)$, but I cannot find anything on this topic.

where $m,n in mathbb{N} $. Does there exist a closed form for this infinite series? What happens if $m = n$? Can we find a closed form when $m = 2, n = 4?$ or when $m = 5, n=6$? What would be a general approach? Any idea? I’ve been working on this infinite series from quite some time.

Hi friends, there is a working instruction on how to make money streaming movies and TV shows. As you know, all video hosting services pay money for views. There are many ways to create a WordPress streaming site, but here’s an alternative way. If everything is set up, then there will be almost complete automation, you just have to monitor the release of new films / TV series episodes and upload them via torrent to your accounts on video hosting.
The engine is on GitHub, the code is all open and completely free, you can make your commits and support the author with an asterisk.

If you want to visually watch the training video, then go to YouTube, but the video is in Russian, turn on subtitles and translate them into English.

I managed to understand everything from the subtitles, so in this one I will give you a transcript of this instruction in English.

Register the domain name of the site (any domain registrar); Buy a Debian or Ubuntu VPS server, RAM > 2GB (any hosting); Add your domain to cloudflare and register ns records with your domain registrar.

Now connect via ssh to the server and install the streaming portal engine to the registered domain. Installation is very simple, you only need to fill in a few fields.

By default, about 1 thousand films from TMDb will be immediately added to the site. You can do even more, you just need to remove the page limitation in the section with the list of films and restart the parsing of the TMDb API. After the end, the number of films and TV series on the site became about 20 thousand. By default, a directory with films and series is created, since it will contain only trailers of films and series. But nothing prohibits starting uploading movies and series to video hosting, the engine will automatically pick them up via API and add them to the site.

I went to a torrent tracker and downloaded the latest episode of the series. Then he started uploading the series through a browser to four video hosting sites. In my city, the Internet is very bad, so downloading one episode at a time is long and painful. But just to take a screenshot, I started uploading the file to all video hosting services through the browser.

Could you please tell me what other video hosting services are there that have an API and FTP upload is available?

After successfully uploading the video file, I added an API line to the top of the list in the admin panel. These are the lines you must add, first replace them with your own API keys:

The episode automatically appeared on my website in the player (the engine checks new episodes by API once an hour).

It is possible to download series for full seasons and send them to several video hosting services from the server at once. Configuration for all FTP video hosting is done over SSH.

1. Bug on ninjastream I didn’t manage to connect to ninjastream the first time, I don’t know what it’s connected with. Failed to create file system for “CINEMATORRENT1:”: NewFs: failed to make FTP connection to “ftp.ninjastream.to:8025”: 550 Checking password error. Therefore, you need to make the connection a second time for CINEMATORRENT1. Enter the same information as the first time. The second time everyone connected successfully.

2. Bug on streamtape Also, it was not possible to upload to streamtape, since for some reason the folders there are not created via FTP. Do you know why? If you find the reason for the error, please write to me.

3. Bug on streamsb And finally, when uploading something to streamsb, sometimes an error appears: Failed to create file system for “CINEMATORRENT3:tt12809988/”: NewFs: failed to make FTP connection to “ftp.streamsb.com:21”: 530 Login incorrect. The data is entered correctly, some series are loading, and some are not, so this is an error on the streamsb side.

I launched a torrent client and accessed it in a browser. Now you can put any movie or TV series on download. I download the whole season of the series. The series files will be in the downloads folder. I find this series on IMDb, copy the identifier, and transfer all the video files with the series to the folder. For the engine to start uploading all episodes one by one to video hosting, you need to move the folder with the ID to the uploads folder. On my server, 1 episode took 1-2 minutes to load. If you have good hosting, load times can be much faster.

After some time (about 1 hour), all episodes of the series automatically appeared on my website in the player.

If you have any more information about working with the engine on your sites, I will be glad to hear the recommendations. Thank you.

Successful streaming everyone

PS: Do not create a streaming portal if it is prohibited in your country. I am located in Malaysia and there is no DMCA in this country.

Let f ∈ C(−π, π). Show that both the nth Fourier sine and the nth Fourier cosine
coefficient of f go to 0 as n → +∞.

Hello guys. I did some work for proof but i am not sure that am i on the right track. I used Bessel’s inequality and got that ||f||^2 ≥ π(Σ(|an|^2 +|bn|^2)
So if i choose f to be zero, then an and bn must be zero. This is my idea but not sure is it true.

I am solving a (infinitely long) differential equation which has the solution $$
y(r)=-frac{c}{5}+frac{l^4c^3}{20r^5}+frac{l^{6}c^5}{16r^9}+mathcal{O}(l^8),
$$
where I am not sure about the sign of the last term. Additionally, I know that this solution is a series of the form $$
y(r)=-left(frac{c}{r}right)cdotp_2F_1(d,e;f;z),
$$
Where $_2F_1$ is a hypergeometric series. My question is: how can I make mathematica find the exact form of the hypergeometric series, based on the first few terms?

I have function $f(x)=frac{1}{3-z}$ which can be rewritten as begin{align}
frac{1}{1-(z-2)} quad text{or}quad frac{1}{3(1-frac{z}{3})}
end{align}
By the $frac{1}{1-r}=1+r+r^2+… $ and $|r|<1$, their Taylor expansion

$frac{1}{1-(z-2)}=sum_{n=0}^{infty} (z-2)^n$ and the radius of convergence $R=1$.

$frac{1}{3(1-frac{z}{3})}=frac{1}{3}sum_{n=0}^{infty}left(frac{z}{3}right)^n$ and radius of convergence $R=3$.

I believe I understand the proof, but I just want to be sure I understand fully what Rudin is saying.

The theorem is:

Suppose $x_n in mathbb{R}^k$ and $x_n = (a_{1,n}, ldots, a_{k,n})$ Then ${x_n}$ converges to $x = (a_1, ldots, a_k)$ if and only if $limlimits_{n to infty}$ if and only if $limlimits_{n to infty} a_{j,n} = a_j$ for $1 leq j leq k$.

The forward direction is more or less clear, though I have only one small question.. Rudin writes that $$
|a_{j,n} – a_j| leq |x_n -x|.
$$
I assume the right side is a vector norm, the right-hand side is the absoltue value of a difference of scalars, because the inequality comes from taking the square root of a square. Is that right?

The backward direction is a bit more confusing. Rudin’s proof, replicated verbatim, is:

Conversely, if (2) holds, then to each $epsilon > 0$ there corresponds an integer $N$ such that $n geq N$ implies $|a_{j,n} – a_j| < frac{epsilon}{sqrt{k}}$. Hence $n geq N$ implies $$
|x_n – x| = left(sumlimits_{j=1}^k |a_{j,n} – a_j|^2 right)^{1/2} < epsilon,
$$
so that $x_n to x$.

Here is my confusion: I think Rudin has skipped a step. He picks an $N$ for only a single $j$, but not for each $j$. It seems to me that we should, for each $j$, pick $N_j$ so that $n geq N_j$ implies $|a_{j,n} – a_j| < frac{epsilon}{sqrt{k}}$ and then set $N = max(N_1, ldots, N_k)$. Otherwise, Rudin is in some way bounding the above sum by a single index, which seems peculiar.

Would I be correct that Rudin has actually done what I just suggested, but been silent about it in the write-up?