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Phulax
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Post Number: 7
Posted on Wednesday, 29 May, 2013 - 02:48 pm:   

I'm working from Bostock & Chandler's 'Further Pure Mathematics', in Misc. ex. 6, Q7 (ii)(b) with ln(r)/r^2 is a real pain, the rest of the question is fine. The answer says (ii)(b) converges but (ii)(a) diverges (which I could show), so that's of no real use. I tried D'Alembert's test but I just get something that tends to 1. I think I need a series with bigger terms so 1/r^2 is of no use either. Any suggestions or alternatives?
lebesgue
Veteran poster

Post Number: 3130
Posted on Wednesday, 29 May, 2013 - 03:09 pm:   

What about convergence of the sum of r^(3/2)? While ln r is increasing, it increases slower than any power. More precisely, given a constant e>0, ln r<r^e for all sufficiently large r. Can you prove something like that? So in particular, with e=1/2, (ln r)/r^2<1/r^3/2 for all sufficiently large r. So if you know that the sum of r^(3/2) is convergent, you are there. By the way, do you know the integral test? You can apply it here as well.
Phulax
Poster

Post Number: 8
Posted on Wednesday, 29 May, 2013 - 03:58 pm:   

Ah right, that was very helpful (I was confused by the use of 'e' at first though!). I didn't know of the integral test, very interesting indeed! I've used differentiation to show ln(x)-sqrtx >0 for all x>1 (dy/dx > 0 => increasing). Thank you for the help.
Phulax
Poster

Post Number: 9
Posted on Wednesday, 29 May, 2013 - 05:13 pm:   

pardon the mistake, I showed sqrtx - ln(x) >0 for all x>1 (not ln(x)-sqrtx >0)

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