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I need assiatance to understand a problem in Inequality

  1. Nov 16, 2011 #1
    Hello Friends,

    I at a loss to understand the parts of the following proof:
    For any positive ineteger n, prove that:

    (1+1/n)^n < (1+1/n+1)^n+1
    a, b positive real numbers such that a < b

    Proof:

    b^n+1 - a^n+1 = (b-a)(b^n+ab^n-1+...+a^n)
    I could not understand the following part:
    By a repeated use of a < b
    (n+1)a^n < (b^n+ab^n-1+...+a^n) < b^n
    .
    .
    .
    How the inequality equation (n+1)a^n < (b^n+ab^n-1+...+a^n) < b^n
    is derived from a < b?
    I can understand how (n+1)a^n < (n+1) b^n, but how (n+1)b^n is greater than (b^n+ab^n-1+...+a^n) and how it's greater than a^n?
    Could anyone please help me?

    Best Regards,
    Sabya
     
  2. jcsd
  3. Nov 16, 2011 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    an < akbn-k < bn for each k, where 0≤k≤n, since a < b. There are n+1 terms in the sum, so the final inequalities hold.
     
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