prove the inequality

Question

0

1073

5

+4624

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

+33654

+2

Best Answer

How about:

.

Alan Oct 19, 2017

Alan · Answer 1 · 2017-10-19T02:10:15

How about:

.

Guest · Answer 2 · 2017-10-18T02:44:13

Cannot prove it formally, but the LHS converges to 2.38423........., which is < 3

Guest · Answer 3 · 2017-10-19T01:39:37

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 · Answer 4 · 2017-10-19T03:40:01

Thank you Alan.