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#1 Re: This is Cool » sums of power sequences » 2019-03-23 22:27:52

cmowla wrote:

I know it's almost been a year since this thread's last post, but I made an "Adjusted Pascal's Triangle" to create these power sum formulas for positive integers, and I finally got around to posting it on this forum.

http://i.minus.com/i67c1SgPQAHxV.png

Which I drew from the following formula I derived "from scratch":

(Putting in the n at the top of the triangle for sum of i^0 worked out perfectly, even though the formula itself cannot compute the correct value for R = 0).

Adjusting this triangle and using the method that knighthawk used in the previous post, I created an "Adjusted Pascal's Triangle" to compute the Bernoulli numbers which can be viewed here (it's too wide to post here, I think).

Basically, I found that

, and I used that to create that "Bernoulli Triangle".

I then wrote the following recursion formula from that Bernoulli triangle.

where

, where you can calculate a Bernoulli number in terms of its predecessor Bernoulli numbers.


Has anyone seen similar images like these "Adjusted Pascal Triangles" to compute the power sum and Bernoulli numbers visually?  I'm curious because I never heard of either, particularly the sum of power one, in high school or college math courses.

http://www.mathisfunforum.com/viewtopic.php?id=23646          same result bro smile .

#3 Re: This is Cool » Partial sum formula of a series by recursion » 2017-01-10 01:32:08

I think that i used the wrong word big_smile (I'm italian), with publish i mean post here. However this is my proof:
FluxBB bbcode test
Knowing that:
FluxBB bbcode test  (excuse me the under subscript is 0)
We can rewrite the series so:
FluxBB bbcode test
Now i notice that delta_n appears m-n times in the series:
FluxBB bbcode test
FluxBB bbcode test
FluxBB bbcode test
FluxBB bbcode test
We must work a little more for our formula:
FluxBB bbcode test
QED

#4 Re: This is Cool » Partial sum formula of a series by recursion » 2017-01-09 23:06:48

Well, I never saw the question by this perspective. I think that my formula is simpler because doesn't need the knowledge of bernoulli numbers that are really hard to remember. And if k gets really high my formula is long because of the recursion but still practicable: i challenge everyone at remembering for a long time a big part of Bernoulli sequence. So should i post the proof or it's a naive work?

#5 This is Cool » Partial sum formula of a series by recursion » 2017-01-09 05:57:13

Eulero
Replies: 5

Hi,
I haven't been here for a while, but now i'm back with something new. I found a formula that give the result of the partial sum of the series:
FluxBB bbcode test
For each k positive integer. With recursion i mean: do you want the partial sum formula for n=3? You need to know the partial sum formula for n=2 and for that you need partial sum formula for n=1;etc.
This is the formula:

FluxBB bbcode test

It works perfectly!
Before I publish the proof i really would like your judge:is it a useful formula? Or it's less interesting than i think?
I thank you for every answer.

#7 Re: Puzzles and Games » An interesting equation » 2016-09-15 07:44:26

Good job!
But the fourth solution you proposed is wrong as you can see here. The exercise exalts one of the most curious proprieties of golden ratio : it's the only not integer number whose himself and reciprocal and square have the same fractionary part. When you have time would you post the execution?

#8 Re: Puzzles and Games » An interesting equation » 2016-09-15 03:16:32

It's a solution but not the only one

#10 Puzzles and Games » An interesting equation » 2016-09-15 02:29:47

Eulero
Replies: 17

Hi,
Today I want to propose you a beautiful problem:
Solve the equation:
mant{x^(-1)}=mant{x}=mant{x^2}
Where mant{x} is the mantissa function

#11 Introductions » Hello from Italy! » 2016-09-14 04:42:04

Eulero
Replies: 6

Hello,
I don't know if there are other italian users, I'm here because Italian Math Forums are really boring because of their inactivity. I'm really young (fifteen) but I have a great passion for math and I really like to solve hard problems and puzzles. I'm sorry if my English isn't perfectly correct , but I'll do my best.
smile

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