Wikipedia:Reference desk/Archives/Mathematics/2021 September 13

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September 13

Sum of integers to negative powers. Finite vs. Infinite

Resolved

Sum (n=1..Inf) n^k is infinite for k=1 and finite for k=2. Where is the boundary? Is it infinite for any k below 2?Naraht (talk) 16:30, 13 September 2021 (UTC)

@Naraht: By the integral test, $\sum _{n=1}^{\infty }n^{-(1+\epsilon )}$ would seem to be finite for all $\epsilon >0$ as $\int _{1}^{\infty }{\frac {1}{x^{1+\epsilon }}}\ dx={\frac {1}{\epsilon }}<\infty$. By the same test, the series is divergent for all $\epsilon \leq 0$.--Jasper Deng (talk) 16:46, 13 September 2021 (UTC)

See Harmonic_series_(mathematics)#p-series, which of course confirms what Jasper said. --Wrongfilter (talk) 16:50, 13 September 2021 (UTC)

OK, so everything strictly between 1 and 2 is finite. k=1.001 gives a finite... Thanx.Naraht (talk) 16:59, 13 September 2021 (UTC)

You can also read Riemann zeta function. Ruslik_Zero 20:11, 15 September 2021 (UTC)