Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #1 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 (I'm italian), with publish i mean post here. However this is my proof:

Knowing that:
(excuse me the under subscript is 0)
We can rewrite the series so:

Now i notice that delta_n appears m-n times in the series:

We must work a little more for our formula:

QED

## #2 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?

## #3 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:
$\sum_{n=1}^{m} n^k$
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:

$\LARGE \sum_{n=1}^{m} n^k = \frac{m^{k+1}+km^{k}- \sum\limits_{n=0}^{m-1} n^{k} \left [ n \left [ \left ( 1+\frac{1}{n} \right ) ^k-1 \right ] -k \right ]}{k+1}$

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.

## #4 Re: Puzzles and Games » An interesting equation » 2016-09-16 03:19:46

Good job zetafunc!

## #5 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?

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

It's a solution but not the only one

Thank you

## #8 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

## #9 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.