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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

**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:

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:

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.

Good job zetafunc!

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?

It's a solution but not the only one

Thank you

**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

**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.

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