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January 5th, 2017, 05:42 AM   #1
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Show that : 1+1/2+1/3+....+1/n>2n/(n+1) for n>1
fahad nasir is offline  
January 5th, 2017, 01:30 PM   #2
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mathematical induction works.


Need to show: (2n+1)/(n+1)>2(n+1)/(n+2)
or $\displaystyle (2n+1)(n+2)>2(n+1)^2$ or $\displaystyle 2n^2+5n+2>2n^2+4n+2$.
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January 5th, 2017, 03:16 PM   #3
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It's a routine induction proof.

Verify the base case, $n = 2$.

Assume it's true for some positive integer $n > 1$.

Then, using that assumption, show that the claim holds for the "$n+1$" case.

Hint: Start by adding $1/(n+1)$ to both sides.
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January 16th, 2017, 09:15 PM   #4
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Originally Posted by fahad nasir View Post
Show that : 1+1/2+1/3+....+1/n>2n/(n+1) for n>1
I use Gauss's idea that he use when he solves the problem '1+2+...+n+...+100'.

A useful inequality: with $a, b>0$, we obtain
$$ \frac{1}{a}+\frac{1}{b} \ge \frac{4}{a+b}.$$

$$\frac{1}{k}+\frac{1}{n+1-k} \ge \frac{4}{n+1}\,\forall k=1, 2, ..., n.$$
$2\sum_{k=1}^{n}\frac{1}{k} \ge \frac{4n}{n+1}.$

Note that: with $a, b>0$,
$ \frac{1}{a}+\frac{1}{b} = \frac{4}{a+b} \iff a=b.$
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Last edited by The Epsilon; January 16th, 2017 at 09:39 PM.
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