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Series and Sequences

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Apply Ratio or Root Test on Absolute value of n-th term to find Interval of ... Eulers Logarithms. Works only for rearrangements of alternating harmonic series ... – PowerPoint PPT presentation

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Title: 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
• 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