Sigma Notation

  1. We can write u1 + u2 + u3 + u4 + ... + un using sigma notation.
  2. Σ is called sigma which is the equivalence of capital S in the Greek alphabet. The first use of Σ as a notation for summation was attributed to Leonhard Euler. [Euler was born in Basle on April 15, 1707. He was widely aknowledged as "Analysis Incarnate."]
  3. n
     
    Σ
    ur = u1 + u2 + u3 + u4 + ... + un
    r =1
     
  4. 4
     
    Σ
    (3k+5) = [3(1)+5] + [3(2)+5] + [3(3)+5] + [3(4)+5]
    k =1
     

    4
     
    Σ
    (3k+5) = [8] + [11] + [14] + [17] = 50
    k =1
     
    Note that this is actually an arithmetic series or summation.
  5. 6
     
    Σ
    (1/3)i = 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729
    i =1
     
    which has a sum of 364/729. Note that this is actually a geometric series or summation. Note that the subcript can be written as r, k or i.
  6. Some Properties of Sigma Notation.
    n
     
    n
     
    n
     
    Σ
    [ ur + vr ] =
    Σ
    ur +
    Σ
    vr
    r =1
     
    r =1
     
    r =1
     

    n
     
    n
     
    Σ
    k ur =
    k Σ
    ur
    r =1
     
    r =1
     
    where k is some constant term.
    n
     
    Σ
    k = kn
    r =1
     
    This is equivalent to adding a constant k, n number of times.
  7.  
    Σ
    ur
    r =1
     
    means summing ur from r=1 to infinity.

Exercises

  1. Expand the following:
    1. 5
       
      Σ
      (2 + 1/r)
      r = 1
       
    2. 4
       
      Σ
      k(3-k)
      k = 1
       
    3. 6
       
      Σ
      5(2)i -1
      i = 1
       
  2. Find the sum of these series:
    1. 200
       
      Σ
      (3r-5)
      r = 1
       
    2. 10
       
      Σ
      7(1.3)k-2
      k = 1
       
    3. 10
       
      Σ
      (2/3)r
      r = 5