3.1 Summation Formulas and Properties

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  General
  • while and for loops

    

  • sequence a1, a2, a3, ..., an
  • finite sum a1 + a2 + ... + an

    

  • infinite sum

    

    which is the same as

    

  • diverges - limit does not exists
  • converges - limit exists
  • absolutely converget series
    • if converges
    • then also converges

      
      
      
      
      
      
      

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  Linearity
  • for constant c and sequences a1, a2, ..., an and b1, b2, ..., bn

    

  • also for infinite series
  • useful to manipulate asymptotic notation.
  • ex:

    

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  Arithmetic Series
  • arithmetic series:

    

  • has value of:

    = (n²)

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  Geometric Series
  • geometric or exponential series
  • for real x 1

    

  • has value of:

    

  • infinite decreasing geometric series
    • infinite summation and |x|<1
    • has values of:

      

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  Harmonic Series
  • nth Harmonic Number for positive integers n:

    

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  Integrating and Differentiating Series
  • additional formulas can be obtained by integrating and differentiating
  • ex:
    • differentiate both sides of

      

    • and then multiply by x, you get

      

      
      
      
      
      
      

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  Telescoping Series
  • for any series a0, a1, a2, ..., an:

    

  • because each term is added and substracted exactly once
  • similarly

    

  • example:

    

    we can write

    

    we get

    

    
    
    
    
    
    
    
    
    

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  Products
  • finite product a1 · a2 · a3 · ... · an

    

  • if n = 0 value is 1
  • product to summation identity: