inequality question (1 Viewer)

[maths lover]

i) show that for x>0, x>Ln(1+x)
ii) hence show that e^(nc2)>n factorial, btw (nc2 is n choose 2) for n=all positive integers.

i already got the first part, i simply tested x=1, and showed that it was an increasing function.
im pretty sure part ii is wrong since n would have a minimum value of 2, however after testing a few different ideas i don't really know where to go with part ii.

 

[Shadowdude]

You have to prove:

Okay, now we note that that is equivalent to:



So if we prove that, we prove what we need to.


We know that:



We use our inequality, . So for example, if we set x = n-1, we know that:

So:



Which is the sum of the first (n-1) integers. We know the sum of the first k integers is:

So:



Which is what you want.

 

[maths lover]

You have to prove:

Okay, now we note that that is equivalent to:



So if we prove that, we prove what we need to.


We know that:



We use our inequality, . So for example, if we set x = n-1, we know that:

So:



Which is the sum of the first (n-1) integers. We know the sum of the first k integers is:

So:



Which is what you want.

thanks for that makes sense. however in your last line of working did u just assume that they were the same expressions by subbing into the calculator or did you somehow prove it.

 

[asianese]

It's just an observation - if you have n(n-1) in the numerator and you want something with n!, the natural thing to do is to times the top by (n-2)(n-3)...(2)(1), and to balance that we times the bottom by that also. The 2nd last bit is noticing that

 

[maths lover]

It's just an observation - if you have n(n-1) in the numerator and you want something with n!, the natural thing to do is to times the top by (n-2)(n-3)...(2)(1), and to balance that we times the bottom by that also. The 2nd last bit is noticing that

yeah i understand now. thanks to all who helped.

 

[lolcakes52]

In part i you can't just test for 1 and say its an increasing function, as it may cross for 0<x<1. You could just show that the inequality holds for the limit as x approaches 0, or that that is when the equality is, then say it is an increasing function.

 

[Shadowdude]

thanks for that makes sense. however in your last line of working did u just assume that they were the same expressions by subbing into the calculator or did you somehow prove it.

I know that



because that's the sum of the first n-1 numbers.

And I know that:



because if you simplify the RHS, you get the LHS

And I know that



because that's like... the definition of it.


So two things you needed to know: Sum of the first k integers, and the 'definition' of the C thing - knowing that