4.3. Special Series

Theorem 4.2.11: p Series

The series is called a p Series.
  • if p > 1 the p-series converges
  • if p 1 the p-series diverges

Examples 4.2.12:
 
  • 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 ?)
Proof:

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.

Now the result follows from the Cauchy Condensation test .

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