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

= September 13 =

Sum of integers to negative powers. Finite vs. Infinite
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)
 * 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), 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)