Posted by Bernie on .
Consider
∞
∑ [(3k+5)/(k²2k)]ᵖ, for each p ∈ ℝ.
k=3
Show this series { converges if p > 1
{ diverges if p ≤ 1
Hint: Determine the known series whose terms past the second give an approximate match for the terms of this series. This series is suitable (almost) for using the comparison test. Separate comparisons with it, or closely related series, are needed to establish convergence or divergence of the series according to p. You will need to establish inequalities, based on approximations (as below), to apply the comparison tests.
k²  2k ≥ k²/2 for k ≥ 4 3k + 5 ≤ 8k
k² > k²  2k k < 3k + 5

calculus 
Bernie,
Those approximations are supposed to look like this:
k²  2k ≥ k²/2 for k ≥ 4
3k + 5 ≤ 8k
k² > k²  2k
k < 3k + 5