Use the inequalities for a suitable value of to show that lies between 3 and .
Hello, roshanhero!
A fascinating problem . . .
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: .
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.