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Math Help - How to prove this by induction ?

  1. #1
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    How to prove this by induction ?

    show that, for n >= 1

    (1+10^-1)(1+10^-2)...(1+10^-n) <=2
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  2. #2
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    Quote Originally Posted by Rakesh View Post
    show that, for n >= 1

    (1+10^-1)(1+10^-2)...(1+10^-n) <=2
    Induction is not suitable for proving this result. The best method is to take logs and use the well-known inequality \ln(1+x)\leqslant x.

    This tells you that \sum_{k=1}^n\ln(1+10^{-k})\leqslant\sum_{k=1}^n10^{-k} = {\textstyle\frac19}(1-10^{-n})<{\textstyle\frac19}. Therefore \prod_{k=1}^n(1+10^{-k})<e^{1/9}<2.
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