ringearV

2021-08-14

The following series $\sum _{k=0}^{\mathrm{\infty }}\frac{1}{{k}^{5}}$ is:
a) alternating series
b) convergent p-series
c) divergent p-series
d) geometric series

Isma Jimenez

$\sum _{k=0}^{\mathrm{\infty }}\frac{1}{{k}^{5}}$
This series is convergent p-series.
Since $\sum _{k=0}^{\mathrm{\infty }}\frac{1}{{k}^{5}}$ converges for $p>1$ and diverges for $p\le 1$
So compare this then: $p=5>1$
Thus, given series converges by p-series test.

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