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Math Help - Show that a sequence is increasing

  1. #1
    Member mohammadfawaz's Avatar
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    Show that a sequence is increasing

    Hello all,

    I need to prove that the sequence c_n = 1+\frac{1}{2}+...+\frac{1}{n} - ln(n+5) is increasing. I did compute c_{n+1}-c_{n} but i couldn't find the sign.
    Help please
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mohammadfawaz View Post
    Hello all,

    I need to prove that the sequence c_n = 1+\frac{1}{2}+...+\frac{1}{n} - ln(n+5) is increasing. I did compute c_{n+1}-c_{n} but i couldn't find the sign.
    Help please
    c_{n+1}-c_{n}=\frac{1}{n+1}-\ln\left(\frac{n+5}{n+6}\right)=\frac{1}{n+1}-\ln\left(1-\frac{1}{n+6}\right)\geqslant\frac{1}{n+1}-\ln\left(1-\frac{1}{n+1}\right).
    It remains to show that x\geqslant\ln(1-x). To see this merely note that f(x)=x-\ln(1-x)\implies f(0)=0,\text{ }f'(x)=1+\frac{1}{1-x}\geqslant 0.
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