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 May 30th, 2019, 09:49 PM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 729 Thanks: 98 Inequality proof Prove : $\displaystyle 1+1/2 +1/3 +....+ 1/2^{k} \geq 1+k/2$. For k=0,1,2,3... .
May 31st, 2019, 04:27 AM   #2
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Quote:
 Originally Posted by idontknow Prove : $\displaystyle 1+1/2 +1/3 +....+ 1/2^{k} \geq 1+k/2$. For k=0,1,2,3... .
shouldn't the series be ...

$\displaystyle 1+1/2 +1/\color{red}{4} +....+ 1/2^{k} \geq 1+k/2$.
For k=0,1,2,3...

 May 31st, 2019, 05:07 AM #3 Global Moderator   Joined: Dec 2006 Posts: 21,029 Thanks: 2259 No. Group the terms like this: 1 + (1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + etc. This sum is at least 1 + (1/2) + (1/4) + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + etc., which is 1 + $k$(1/2). Thanks from topsquark and idontknow

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