Thread: a number logic probem

It is a major theorem that



Thanks!

Hello, soyeahiknow!

It is a major theorem that:

. . $\displaystyle \displaystyle{S \;=\; \sum^{\infty}_{n=1}\frac{1}{n^2} \;=\;\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \hdots \;=\;\frac{\pi^2}{6} }$

Use this to calculate: .$\displaystyle [1]\;\dfrac{1}{2^2} + \dfrac{1}{4^2} + \dfrac{1}{6^2} + \hdots$

We have: .$\displaystyle X \;=\;\dfrac{1}{2^2} + \dfrac{1}{4^2} + \dfrac{1}{6^2} + \hdots $

. . . . . . . .$\displaystyle X \;=\;\dfrac{1}{2^2}\underbrace{\bigg[\dfrac{1}{1^2} + \dfrac{1}{2^2} + \dfrac{1}{3^2} + \hdots \bigg]}_{\text{This is }S} $

. . . . . . . .$\displaystyle X \;=\;\dfrac{1}{4}\cdot\dfrac{\pi^2}{6}$

. . . . . . . .$\displaystyle X \;=\;\dfrac{\pi^2}{24}$

$\displaystyle [2]\;\dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{5^2} + \hdots$

Are the denominators primes? .$\displaystyle 2, 3, 5, 7, 11, 13,\:\hdots$

or a Fibonacci-type sequence? .$\displaystyle 2, 3, 5, 8, 13, 21,\:\hdots$

I believe they are primes

can u tell me why you took out the 1/(2^2) ?

Originally Posted by Soroban

Hello, soyeahiknow!


We have: .$\displaystyle X \;=\;\dfrac{1}{2^2} + \dfrac{1}{4^2} + \dfrac{1}{6^2} + \hdots $

. . . . . . . .$\displaystyle X \;=\;\dfrac{1}{2^2}\underbrace{\bigg[\dfrac{1}{1^2} + \dfrac{1}{2^2} + \dfrac{1}{3^2} + \hdots \bigg]}_{\text{This is }S} $

. . . . . . . .$\displaystyle X \;=\;\dfrac{1}{4}\cdot\dfrac{\pi^2}{6}$

. . . . . . . .$\displaystyle X \;=\;\dfrac{\pi^2}{24}$




Are the denominators primes? .$\displaystyle 2, 3, 5, 7, 11, 13,\:\hdots$

or a Fibonacci-type sequence? .$\displaystyle 2, 3, 5, 8, 13, 21,\:\hdots$

I think the second one should be the sum of the squared inverses of the odd natural numbers, otherwise it seems to be pretty hard to calculate the sum over the primes...

Tonio

Originally Posted by tonio

I think the second one should be the sum of the squared inverses of the odd natural numbers, otherwise it seems to be pretty hard to calculate the sum over the primes...

Tonio

Prime Zeta Function -- from Wolfram MathWorld

id:A085548 - OEIS Search Results

EDIT: I should have added some comment to be clear; it's likely that tonio is right that the problem is misstated; nevertheless, if it is correctly stated then the above links are relevant.

Originally Posted by undefined

Prime Zeta Function -- from Wolfram MathWorld

id:A085548 - OEIS Search Results

Thanx, I knew that, but I highly doubt an obviously beginner student can come up with a research about the prime zeta function...

Tonio

Originally Posted by tonio

Thanx, I knew that, but I highly doubt an obviously beginner student can come up with a research about the prime zeta function...

Tonio

Yes, I edited my post above with a comment, but not quick enough.

wait wait,

i think there may have been a mistake.

The 2nd problem is just simply:

1/(1^2) + 1(3^2) + 1/(5^2)

So it is simply the odd numbers. So would I just do 1/(3^2) * S= 1/9 * (Pi^2)/6 = (Pi^2)/ 54? Thanks

Originally Posted by soyeahiknow

wait wait,

i think there may have been a mistake.

The 2nd problem is just simply:

1/(1^2) + 1(3^2) + 1/(5^2)

So it is simply the odd numbers. So would I just do 1/(3^2) * S= 1/9 * (Pi^2)/6 = (Pi^2)/ 54? Thanks

In post #2, Soroban factored out (1/2)^2 because it evenly divides each term, and because doing so allows us to express the sum as a product of already known quantities.

For this problem, think about what happens when you add this series to the series that Soroban solved.

if I add this series to the one Soroban solves, it would just be back to the original series right?


OOOOO so all I have to do is ( original series - series with all the even numbers)= series for problem 2 ( the odd numbers). So (pi^2)/6-(pi^2)/24= (pi^2)/8

and what kind of problems are these called? Thanks so much and sorry for the typo in the beginning! I posted it at like 3 am.

Originally Posted by soyeahiknow

if I add this series to the one Soroban solves, it would just be back to the original series right?


OOOOO so all I have to do is ( original series - series with all the even numbers)= series for problem 2 ( the odd numbers). So (pi^2)/6-(pi^2)/24= (pi^2)/8

and what kind of problems are these called? Thanks so much and sorry for the typo in the beginning! I posted it at like 3 am.

You're right about the sum of the squared inverses of the odd naturals, and these series could be classified as ones related to the Riemann zeta function, a truly fascinating and intriguing series.

Tonio