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Prove the inequality:

\(\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^{2}}\right)\ldots\left(1+\frac{1}{2^{n}}\right)<3.\)

tertre  Oct 17, 2017

Best Answer 

 #4
avatar+26752 
+2

How about:

 

.

Alan  Oct 19, 2017
 #1
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Cannot prove it formally, but the LHS converges to 2.38423........., which is < 3

Guest Oct 18, 2017
 #2
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Can you prove that the LHS converges to 2.38423 ?

Melody  Oct 19, 2017
 #3
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As I said "Cannot prove it formally", but it does converge to 2.38423..........

https://www.wolframalpha.com/input/?i=%E2%88%8F+%5B(1%2B2%5E-n),+n,+1,+1000%5D

∏ [(1+2^-n), n, 1, 1000] ≈2.384231029031371724149899288678397238772...............etc.

Guest Oct 19, 2017
 #4
avatar+26752 
+2
Best Answer

How about:

 

.

Alan  Oct 19, 2017
 #5
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Thank you Alan.

Guest Oct 19, 2017

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