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Rudin 1.21

  1. Sep 4, 2008 #1
    In Rudin 1.21 he says the following in the midst of proving a theorem,

    "The identity b[tex]^{n}[/tex] - a[tex]^{n}[/tex]= (b-a)(b[tex]^{n-1}[/tex] + b[tex]^{n-2}[/tex]a + ... + a[tex]^{n-1}[/tex]) yields the inequality

    b[tex]^{n}[/tex] - a[tex]^{n}[/tex] < (b-a)nb[tex]^{n-1}[/tex] when 0 < a < b"

    I can understand that it is less than, but I cannot understand how it is coming (yielding) from the identity.

    Any explanation would be greatly appreciated.
     
  2. jcsd
  3. Sep 4, 2008 #2

    Vid

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    Try seeing what happens to the second term on the right side when b=a.
     
  4. Sep 5, 2008 #3
    Vid's point is that:

    [tex]b^n-a^n=(b-a)(b^{n-1}+b^{n-2}a+.....+ba^{n-2}+a^{n-1})<(b-a)\underbrace{(b^{n-1}+b^{n-2}b+.....+bb^{n-2}+b^{n-1})}=(b-a)nb^{n-1}[/tex] since a<b
     
  5. Sep 5, 2008 #4
    ...that's clever. Thank you.
     
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