Wikipedia:Reference desk/Archives/Mathematics/2024 February 10

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February 10

Sum n =1->Inf of 1/(n^p)

I know that this sum is infinite if p is 1 and finite if p = 2, but where is the crossing line. Is it finite or infinite for 1.1? 1.5? 1.9? Naraht (talk) 02:35, 10 February 2024 (UTC)

It's finite for all p > 1, and infinite at p = 1. Wikipedia:Reference desk/Archives/Mathematics/2023 November 1#Border between finite and infinite has some more useful information. GalacticShoe (talk) 02:47, 10 February 2024 (UTC)

Thank you!Naraht (talk) 03:08, 10 February 2024 (UTC)

@Naraht: You should also know that this function has a name. It's called the Riemann zeta function. More precisely, $\sum _{n=1}^{\infty }{\frac {1}{n^{p}}}:=\zeta (p)$, where $\zeta$ is the function in question. Duckmather (talk) 02:40, 13 February 2024 (UTC)

The more common use of the colon equals symbol in a definition is to put the definiendum on the lhs and the definiens on the rhs:^[1]

$\zeta (p):=\sum _{n=1}^{\infty }{\frac {1}{n^{p}}}.$

Since the symbol is asymmetric, one might use an equals colon for the reverse defining direction:

$\sum _{n=1}^{\infty }{\frac {1}{n^{p}}}=:\zeta (p).$

However, while readily understandable, this is not common usage.  --Lambiam 10:07, 13 February 2024 (UTC)