Series and Sequences

1
Series and Sequences

• What and What Not to Do
• Gordon Ivanoski

2
Positive Terms

• Comparison test
• Limit Comparison Test
• Integral Test
• Ratio Test
• Root Test

3
Alternating Series

• N-th term approaches 0
• Strictly decreasing
• Strictly alternating

4
Power Series

• Apply Ratio or Root Test on Absolute value of
  n-th term to find Interval of Convergence
• Check Endpoints

5
Remember

• Just because the n-th term goes to zero doesnt
  necessarily mean the series converges

6
Classic Divergent Series

• Harmonic series
• Geometric Series where R is greater than one
• Compare with known divergent series

7
Eulers Logarithms

• Works only for rearrangements of alternating
  harmonic series
• Many times involves layering

8
Important Series to Remember

• Sine and cosine
• Ln(1x)
• E-series
• Arctan x
• Geometric series where R is less than 1

9
Make Your Own Series

• Integrate or differentiate known power series
• Multiply existing series
• Replace with other variables

10
Exotic Methods

• Kummers test
• Raabes test
• Gausss test

11
Taylors Theorem With Remainder

• Gives an interval for an n-th term and an error
  estimate
• Lagranges formula