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

= 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. Reference desk/Archives/Mathematics/2023 November 1 has some more useful information. GalacticShoe (talk) 02:47, 10 February 2024 (UTC)
 * Thank you!Naraht (talk) 03:08, 10 February 2024 (UTC)
 * 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:
 * $$\zeta(p):=\sum_{n=1}^{\infty}\frac1{n^p}.$$
 * Since the symbol is asymmetric, one might use an equals colon for the reverse defining direction:
 * $$\sum_{n=1}^{\infty}\frac1{n^p}=:\zeta(p).$$
 * However, while readily understandable, this is not common usage. --Lambiam 10:07, 13 February 2024 (UTC)