Direct Comparison Test

Question

Use the Direct Comparison Test to determine if each series converges or diverges.

∑ (Infinity)(n=1) (√(n+4)/(n^4+4))

My book is showing that it compares to √5 ∑ (Infinity)(n=1) (1/n^(3/2)). I am just confused as to where the √5 came from.

Answer 1

Since √(n⁴+4) > √n⁴, 1 / √(n⁴+4) < 1/ √n⁴

{n+4} = {5, 6, 7, ...} Note that 6 < 5(2), 7 < 5(3), etc. So, √(n+4) ≤ √5√n

Therefore, √[(n+4) /(n⁴+4) < √5√n / √n⁴ = √5(1/n^3/2)

Since the "larger" series, √5∑(1/n^3/2), is a multiple of a convergent p-series, the "smaller" given series also converges by the Direct Comparison Test.