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Corollary: p Series

The series 09_infinite_series-220.gif, where p is constant, is called the p series.

COROLLARY

The p series 09_infinite_series-221.gif converges if p > 1 and diverges if p ≤ 1.

PROOF

Case 1: p = 1.

The p series is just 09_infinite_series-222.gif

Case 2: p > 1.

The improper integral converges,

09_infinite_series-223.gif

Therefore the p series converges.

Case 3: p < 1.

The improper integral diverges to ∞, if

(1/xp) dx = limu→∞ ∫1u x-p dx = limu→∞ (u1-p - 1)/(1 - p) = ∞.

Therefore the p series diverges to ∞.
Home Infinite Series Series With Positive Terms Corollary: p Series

Last Update: 2006-11-07