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Recursive defintion of the product notation

  1. Jul 13, 2007 #1
    I'm asked to define recursively (definition by induction) [itex]\prod_{k=1}^{n}a_k[/itex]

    Well, I wrote down the following:

    [tex]\prod_{k=1}^{1}a_k = a_1[/tex]

    Assuming [itex]\prod_{k=1}^{n}a_k[/itex] has been defined for some [itex]n\geq1[/itex],

    [tex]\prod_{k=1}^{n+1}a_k = a_{n+1}\prod_{k=1}^{n}a_k[/tex]

    A similar method was used to define the summation notation in the text, so I used it here.

    But the answer given at the back is

    [tex]\prod_{k=1}^{0}a_k = 1; \prod_{k=1}^{n+1}a_k = a_{n+1}\prod_{k=1}^{n}a_k[/tex]

    I don't understand why the index goes from 1 to 0, and why they have defined it to be 1. Please clarify this.
     
  2. jcsd
  3. Jul 13, 2007 #2

    mathman

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    It is a somewhat pedantic point. When the upper limit goes from n=0 to n=1 (induction step), you need 1 for the n=0 product to get a1 for the n=1 product.
     
  4. Jul 13, 2007 #3
    The reason they define the product starting from k=1 to 0 is to define the empty product to be 1. This is a useful definition, as it avoids many special cases. It is clearly an equivalent definition to yours for all upper bounds greater than 0, since their inductive step is the same and by their definition,

    [tex]\prod_{k = 1}^1 a_k = a_1 \prod_{k = 1}^0 a_k = a_1 \cdot 1 = a_1.[/tex]
     
  5. Jul 14, 2007 #4
    Thanks for the replies. Could you give me a simple example where this definition will be useful? Also, why was not the summation notation defined to be [itex]\sum_{1}^{0} = 0[/itex]? [As implied in my first post, [itex]a_1[/itex] was defined as the sum from 1 to 1.]
     
  6. Jul 14, 2007 #5

    mathman

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    In summation notation sum from 1 to 0 is considered 0.
     
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