Month: March 2022

I found another cool formula for the odd sums

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I noticed that if you put in the sums of natural numbers raised to odd powers you get a recurring polynomial:

For sum of 13:

So every sum with odd powers consists of a polynomial with a building block the simple sum of just the natural numbers to n:

So in general you can write:

Where you can find the coefficients a_k like this:

Where the columns are the coefficients and the rows the corresponding powers.

It is actually kind of an interference pattern, we know that there are no odd exponents in the expanded polynomial so we can set those equal to zero. And we know that the almost last one is equal to \frac{1}{2} so that is enough. There are more equations we know but we don’t even need those.

The entries are the \binom{j+1}{i-2} entries. I’ll add later the best formula

To find the coefficients a_k you have to solve the following matrix

If you an proof that the last one always equals 1/2 and the others are 0 then you completely proved this whole formula, that seems pretty simple actually.

Would there be anything like it for the even powers?

Maybe there is, look here:

There is also one hidden in k^8

EDIT:

I found a similar formula for the even numbers:

Relation between regular sum and sum of cubes.

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\sum_{k=1}^n k = \frac{n(n+1)}{2}

\sum_{k=1}^n k^3 = \frac{n^2(n+1)^2}{4}

This looks really similar! In fact it looks like the sum for the cubes is just the regular sum squared? This must be a simple way to prove the third formula it seems! I looked in to it and it goes like this:

Lets take for example the simple sum of \sum_{k=1}^4 k = 1+2+3+4 = 10

( \sum_{k=1}^n k )^2 = (\frac{n(n+1)}{2})^2 = \frac{n^2(n+1)^2}{4} = \sum_{k=1}^n k^3

(\sum_{k=1}^4 k )^2 = (1+2+3+4)(1+2+3+4) = 1^3 + 2^3 + 3^3 + 4^3 = \sum_{k=1}^4 k^3

If we make a table of the following multiplication (1+2+3+4)(1+2+3+4)

We find:

1234
11^22*13*14*1
22*12^23*24*2
33*13*23^24*3
44*14*24*34^2

Now to find the sum we just have to sum all the entries in the table above. The insight is to look at the entries of the squares and that look at the entries to the left and upwards

So we find a sum of squares, but we need to convert this to a sum of cubes somehow
Lets pair every squared term with its neighboring blue entries and look if we can find a pattern
So the squared term of 4 and both blue barred entries that have the factor 4 sum to 4 cubed!
In general for the n’th square. So that means if we take the squared sum that we will end up with sum of cubes

This pattern probably repeats in the \sum_{k=1}^n k^m, m\in \mathbb{N}

There is a similar pattern for the squared sum of cubes and the sum of the fifth power. There is probably a simpler way to prove this. Maybe involving the Cauchy product of sums

The cry of the non-native English speaker in the wild (internet)

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Before I begin my actual comment, I would like to apologize in advance for my inadequate level of English proficiency. I am not a native speaker of the world’s current lingua franca which unfortunately leads to me making numerous embarrassing mistakes being made whenever I attempt to communicate using this language. Whenever I am reminded of how I lack the ability to convey my thoughts in an eloquent manner I feel as though I have committed a cardinal sin, as though every English teacher in the world are simultaneously shaking their heads and sighing due to how utterly disappointed they are at me.

Although I know that saying sorry to those of you who are reading my comment will not change the fact that I fail miserably to write and speak perfect English, I am writing this as a way to deter a certain type of people who cannot stand poor English (Also known informally as “Grammar Nazis”) from mocking me by posting unwanted and unnecessary comments detailing my every blunder. In my humble opinion, making grammatical errors should be perfectly acceptable as native speakers should not expect non-native speakers to be able to communicate in their second or third languages eloquently. If you are able to completely understand what the other person wrote, is there really a problem with what they’ve written? No, because the entire concept of communication is the exchange of information between other intelligent beings, which means that no matter how the exchange of information is made, as long as the information is accurately shared there is not a fundamental issue with their ability to communicate. To see it in another way, remember that someone who isn’t fluent in English is fluent in another language. When you think about it this way, isn’t it impressive for someone to speak a second language in any capacity? Having empathy and respect are qualities that are sorely missing for far too many people these days, especially on the internet.

That being said, I am aware that not all netizens who correct others are doing it to ridicule and shame. There are some who do so with the intent to help others improve and grow. However, displaying the failures of other people publicly will cause the person who is criticized to feel negative emotions such as shame and sadness due to the fact that their mistake has been made obvious which severely undermines the point they were trying to make in spite of their unfamiliarity with the English language. In most circumstances people are not looking for language help when they post anything online. Most people just want to enjoy themselves and have a good time on the internet which is why I would not encourage correcting other people regardless of your intentions. If you really do want to help others with their spelling or grammar, I would highly recommend you to help via messaging privately because not only will you not embarrass anyone, you can also go more in-depth with your explanation which I’m sure the other person will greatly appreciate if they want help, but I digress. I know that I’ve written a bit of an essay, but I hope I’ve made my points clear. Anyways, here is the comment I wanted to make:

Lol

Mask phenomena

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The COVID pandemic brought a lot of phenomena that weren’t directly related to the virus but was something no one ever really experienced.

A very common thing was because of the widespread mask usage that half of the faces of people were hidden. Since the face and facial movements is extremely important in social interactions, these interactions were quite different for people wearing masks and not. Some people have a lot of good features hidden by a mask and vice versa. So for a proportion of people the interactions changed if they had their mask on and not. This was quite strongly for me. It was black and white the difference and sometimes if I removed my mask I could see the change on their face actual happening. I work in the hospital so this was quite common for me.

The same is for online interactions and in real life. I wonder if there is research to this.

Neurological immune response based in actual conscious memory of illness

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If you get ill from infection you get certain symptoms (cough, chest pain, painful toes, burning skin). If you get an infection from a certain pathogen (e.g. s. Aureus) you will get similar symptoms then before. Patients can consciously recognize similar symptoms so they will know that they have a recurrent infection. What if this memory response based on similar symptoms that happens consciously also primes the immune system unconsciously to the infection? This would mean that there is a link between the hippocampus and the immune system. Because of the extremelu high evolutionary infectious pressure and similar conmecrions betweem the central nervous system and the immune system (n. Vagis) I think there has to exist a connection like this