Hint:
spoiler
The size of a set is the number of possible values that an element can take.
[2024/06/28] Approximate log base 2
I recently started reading TAOCP, in other words you can expect daily posts from me again, because I'll just take some of the cooler questions from there and repost them here.
because I have never heard of this argument before, ever. most media's stance on politics is "their party bad our party good", but the "all the parties are pretty hypocritical" argument has never been explored properly, because its depressing and nobody likes it.
I've learnt (read text in post)
I'm a Londoner, I used to have this friend (who is not a Londoner) we had a huge disagreement on topic unspecified. But after I've watched this video I think I see his viewpoint, which is true. I just don't see it at all because there's such a enormous disconnect between London and the rest of the country.
I would recommend you to watch the video as well, some arguments made in the video are slightly misleading, but the general picture is clear and true.
https://youtu.be/b5aJ-57_YsQ
yup thats the intended solution, im not really familiar with taylor series yet, but maybe for a person who knows taylor series would be able to see it right away
Hint
spoiler
The solution I have in mind is related to the Taylor series
Hint 2
spoiler
It converges to -ln(2), but why
Solution:
Here's a rly cool solution from stackexchange, which blows my average geometric solution out of the water
spoiler
I've shown that ln(n/n-1) is always larger than 1/n, so Σln(n/n-1) for all natural number n will be larger than the series 1+1/2+1/3+...
but I don't know how to make sure the sum of all ln(p/p-1) only when p is prime is larger than the provided series
the question is strongly suggesting its divergent, i just dont know how to show it
[2024/05/25] Angle between two vectors
- Show that
cosθ=(u⋅v)/(|u||v|)for 2D vectors u and v.
(it is quite hard to come up with these challenges, so if you got any ideas, please post them)
i pulled the image from a meme channel, so i dont know if its real or not, but at the same time, this below does look like a legit response
the background it likely ai generated anyways
(i took the meme off some discord channel, so i dont know how its made)
Hint:
spoiler
It is not a telescoping series
Solution:
i thought the "default" counter example is y=|x| lol
i got the answers, but i dont really know why
Solution (starter question):
spoiler
Please refer to the main post, if you don't like looking at the image. https://gmtex.siri.sh/fs/1/School/Extra/Maths/Unsolved/1d-gravity.html
For the main question, you are encouraged to share your progress
spoiler
You might be able to solve this with differential equations, or by solving the iterative functions, I dont know
[Unsolved] [2024/05/17] 1 dimensional gravity
I've even got a starter question to get you guys into the scenario.
Once you've completed the starter question, under the solution comment attaches the main question, which is unsolved.
i added the solution to the post, i didnt see the multiplication before someone mentioned it, but yeah if we put it to the power of e it will telescope again, which is clearly the main character of this sub at this point (jk)
Hint:
spoiler
e
Solution:
spoiler
zkfcfbzr solved it
i put everything into ln because i was scared of multiplication
[2024/05/16] Infinite multiplication series
- Show that the infinite multiplication
(1+1/1)(1+1/2)(1+1/3)...does not converge.
i showed the question to my friend who isnt particularly bright in maths, he said 30 by just looking at it, i freaked out a bit but it might just have been a lucky guess
[2024/05/15] Differentiability implies continuity
- Show that if a function is differentiable for an interval, it is continuous over that interval.
- A function is continuous if lim_x->a f(x) = f(a)
[2024/05/14] General differential equation
- Express
yin terms ofxfor differential equationdy/dx=ylny
(I'm officially out of ideas again)
[2024/05/13] Irrational powers
- Show that it's possible
a^b=cwhereaandbare irrational, andcis rational.
Sry for the gap I ran out of ideas.
A Brief History of LaTeX - How it became so popular.
Have you seen this font before? This is Computer Modern and it's used everywhere from exams sheets to research papers, all because it is default font in popular typesetting language LaTeX, known for its ability to diplay maths equations, as well as being the de facto standard of writing articles in many fields of science.
Which is surprising as LaTeX is a far less productive option compared to office suites, according to this study, it is much easier to make mistakes in LaTeX than word, something you don't want on a research paper. So how did it against all odds, become the favourite of scientific communities?
Typesetting is the art of placing words on a page. Back in the 70s, it was usually done manually by professionals. However, this was not an option for Donald Knuth, as his publisher was too broke to afford one, instead they used computers to typeset his book, to which he was deeply dissatisfied with the poor results. Believing he can do better, he set out to create his own typesetting system for anyone to produce high-quality books with minimal effort.
This was no easy feat, as the system will have to know the dimensions of all characters to find the optimal arrangement of characters per line, and lines per page. It took him 7 years of typography, and by 1978, he released the first version of TeX along with the Computer Modern font.
After the initial learning curve, TeX slowly gained popularity among researchers. Soon words reached the American Math Society, who at the time was looking for a good digital typesetter. They quickly realised that TeX was exactly what they were looking for. Not only did they heavily promoted the use of TeX to members. Instead of waiting for new features to be added, they went ahead and extended TeX's ability to typeset maths using its powerful macro system, publishing the first ever macro package - AMSTeX.
At the time TeX was able to produce high quality publications, but it was still very complicated to use. To make TeX accessible to everyone, 1984 LaTeX was created so macro packages such as AMSTeX can be used easily. This variant of TeX was so popular by 1990, it became the de facto standard in the scientific community.
Since then LaTeX has been adopted by students for note taking, some markdown editors even include it as a proper way to display maths equations. On the flip side, vanilla LaTeX is still the favourite of many experienced users. And as I see it, there is nothing more fun than perfecting your notes to the point people like it more than the textbook.
So have you hear of LaTeX before? What do you use LaTeX for? Let me know in the comments down below, anyways I'll be seeing you in two weeks, have a good one.