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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) reply

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

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

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) reply

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