# ^^∑ 1/n² = 1 + 1/2² + 1/3² + ...   Basel problem.

 1 + 1 2 + 13 + 14 + 15 + ... ≡ ∑n≥1 1n serie armonica 1 + 1 22 + 132 + 142 + 152 + ... ≡ ∑n≥1 1n2 ∑ 1/n²  Basel problem 1 + 1 2s + 13s + 14s + 15s + ... ≡ ∑n≥1 1ns ζ(s) Zeta function

∑ 1/n² , named after Basel, hometown of

• Bernoulli family who unsuccessfully attacked the problem
• Euler who solved it, and brought him immediate fame when he was 28.

## Teo: ∑ 1/n² e' una serie convergente

dim: minorante della serie dei reciproci dei nr triangolari

 1 + 1 22 + 132 + 142 + 152 + ... ≡ ∑n≥1 1n2 2 1*2 + 2 2*3 + 2 3*4 + 2 4*5 + 2 5*6 + ... ≡ ∑n≥1 2n(n+1)

## Teo:

 1 22 · ∑n≥1 1n2 = ∑n pari 1n2

 ∑n≥1 1n2 − ∑n pari 1n2 = ∑n dispari 1n2

 ∑n≥1 1n2 − ∑n pari 1n2 − ∑n pari 1n2 = ∑segni 1n2 alterni

### Forma sintetica, sottindendendo gli addendi

 1 22 ∑ = ∑pari

 ∑ − ∑pari = ∑dispari

 ∑ − ∑pari − ∑pari = ∑segni alterni

To lissen is an effort, and just to hear is no merit. A duck hears also.

- Igor Stravinsky

# Talk

 ∑ 1n2 - ∑ 1n2 - ∑ 1n2 = ∑ 1n2 n≥1 n pari n pari segni alterni