\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{5}}{{\left({k}+{4}\right)}^{{4}}}\)

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.

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.