Use the inequalities for a suitable value of to show that lies between 3 and .

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- Feb 10th 2010, 10:10 AMroshanheroVery hard inequality
Use the inequalities for a suitable value of to show that lies between 3 and .

- Feb 10th 2010, 11:22 AMJester
- Feb 10th 2010, 11:23 AMSoroban
Hello, roshanhero!

A fascinating problem . . .

Quote:

Use the inequalities: for a suitable value of

to show that lies between 3 and .

We have: .

is an increasing function on the interval

. . Then: .

Let

We have: .

. . Hence: .

We have: .

is an increasing function on the interval

. . Then: .

Let

We have: .

. . Hence: .

Therefore: .

- Feb 10th 2010, 11:26 AMTWiX
What about using the calculator ???

- Feb 10th 2010, 11:49 AMroshanhero
How can we show that the inequality holds for all values of greater than 0.

- Feb 10th 2010, 12:12 PMDrexel28
There are many ways. For example, the assertion is clear if so assume . We know that using some clever manipulation of the series, and using the fact that we may conclude that .

Alternatively, the claim that is equivalent to . Since an integral is positive precisely when it's integrand is positive over the region of integration all we need to show is that but, this follows readilly.

Lastly, the easiest way is to just note that if then and so .

Or, actually. You could use the fact that is Lipschitz (with Lipschitz constant one) to conclude that since . Which, is a stronger claim.