Lecture Notes: Power Series and p-Series

Introduction to Power Series

• Power series (also called p-series) are series of the form ( \frac{1}{n^p} ).
• Examples: ( 1 + \frac{1}{2^p} + \frac{1}{3^p} + \ldots ), where ( p > 0 ).

Special Case: Harmonic Series

• When ( p = 1 ), this becomes the harmonic series:
  • ( 1 + \frac{1}{2} + \frac{1}{3} + \ldots )
• The harmonic series diverges, proven using the integral test:
  • Set up improper integral, which diverges, implying series divergence.

General Case: Power Series with ( p \neq 1 )

• Use the integral test for ( f(x) = \frac{1}{x^p} ).
• Conditions: ( f(x) ) is continuous, positive, decreasing for ( x \geq 1 ).
• Convert ( \frac{1}{x^p} ) to ( x^{-p} ) for integration.

Power Rule for Integration

• Apply the limit as ( a \to \infty ):
  • Add 1 to the power, multiply by reciprocal of new power.
  • Expression: ( \frac{1}{-p+1} \times \lim_{a \to \infty} (x^{-p+1}) ).
• Convergence criteria:
  • If ( -p+1 < 0 ), integral converges.
  • If ( -p+1 > 0 ), integral diverges.
• Conclusion: For series convergence, ( p > 1
  • If ( p < 1 ), series diverges.
• Harmonic series diverges when ( p = 1 ).

Applying Series Tests

• Identify p-series and distinguish from geometric series.
• Use various tests:
  • Geometric series test
  • Telescoping test
  • nth term test
  • Integral test
  • p-series test

Examples and Applications

• Example 1: ( 1 / (2n - 1) )

  • Not geometric, telescoping, or p-series.
  • Integral test shows it diverges due to limit approaching infinity.

• Example 2: ( 3 \times \sum \frac{1}{n^{5/3}} )

  • p-series with ( p = 5/3 > 1 ) so it converges.
  • Factor out constant (3) from series.

• Example 3: ( 1 / n^{\pi} )

  • p-series with ( p = \pi > 1 ) so it converges.

• Example 4: Factorials over numbers ( \frac{n!}{n^n} )

  • Use nth term test; factorials grow faster than powers.
  • Series diverges as limit does not equal zero.

Conclusion

• Remember the different tests and their applicable conditions.
• Practice identifying series types and applying appropriate tests.
• More tests will be introduced, but ensure understanding of current tests: geometric, telescoping, nth term, integral, and p-series tests.