Homework Help: calculus

Posted by Bernie on Saturday, August 6, 2011 at 10:58am.

Consider
∞
∑ [(3k+5)/(k²-2k)]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, Saturday, August 6, 2011 at 11:00am
Those approximations are supposed to look like this:

k² - 2k ≥ k²/2 for k ≥ 4

3k + 5 ≤ 8k

k² > k² - 2k

k < 3k + 5

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