Tell whether the series converges or diverges. sum_(k = 1)^oo 5/(k + 4)^4

slaggingV 2021-08-12 Answered
Tell whether the series converges or diverges.
\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{5}}{{\left({k}+{4}\right)}^{{4}}}\)

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Expert Answer

Lacey-May Snyder
Answered 2021-08-13 Author has 25559 answers
\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{5}}{{\left({k}+{4}\right)}^{{4}}}\)
Put \(\displaystyle{k}+{4}\Rightarrow{n}\)
So, \(\displaystyle{\sum_{{{n}={5}}}^{\infty}}\frac{{5}}{{n}^{{4}}}\)
From p-series test:
For series \(\displaystyle{\sum_{{{n}={1}}}^{\infty}}\frac{{1}}{{n}^{{p}}}\), if p> 1, series converges.
So, for series in equation 1
P = 4, so this series converges
The series is p-series with p = 4 and it converges.
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