p Series Test

• if p > 1 the p-series converges
• if p 1 the p-series diverges

Examples:

• Does the series converge or diverge ?
• Does the series converge or diverge ?
• Does the series converge or diverge ? (This is the same series as in the example for the Limit Comparison test . Are we running in a circle here ?)

• If p < 0 then the sequence converges to infinity. Hence, the series diverges by the Divergence Test.
• If p > 0 then consider the series = . The right hand series is now a Geometric Series.
  • if 0 < p 1 then 2 ^1-p 1, hence the right-hand series diverges
  • if 1 < p then 2 ^1-p < 1, hence the right-hand series converges
• Now the result follows from the Cauchy Condensation test .