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May 30th, 2019, 09:49 PM   #1
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Inequality proof

Prove : $\displaystyle 1+1/2 +1/3 +....+ 1/2^{k} \geq 1+k/2$.
For k=0,1,2,3... .
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May 31st, 2019, 04:27 AM   #2
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Quote:
Originally Posted by idontknow View Post
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...
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May 31st, 2019, 05:07 AM   #3
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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).
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