The following series ∑k=0∞1k5 is: a) alternating series b) convergent p-series c) divergent p-series d)...

ringearV

Answered question

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

Answer & Explanation

Isma Jimenez

Skilled2021-08-15Added 84 answers

$\sum _{k=0}^{\mathrm{\infty}}\frac{1}{{k}^{5}}$
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.