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Title:

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
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
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