limit questions

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April 11th 2011, 09:48 AM
poirot

limit questions

Find the radius of convergence of the power series: $((n!)^2/(2n)!)x^n$

My answer; 0 using $a(n+1)/a(n)$

Find the limits, if they exists, of
i) $100(ln(x))/x^3$ as n tends to infinity

ii) $(sinx-x)/(cosx -x^2-1)$ as n tends to 0

My working i) pretty certain its infinity but not sure how to show it

ii) no idea on this one.
April 11th 2011, 11:02 AM
Opalg

Quote:

Originally Posted by poirot

Find the radius of convergence of the power series: $((n!)^2/(2n)!)x^n$

My answer; 0 using $a(n+1)/a(n)$

$a(n+1) = \dfrac{((n+1)!)^2}{(2(n+1))!}.$ But $(n+1)! = (n+1)n!$ , and $(2(n+1))! = (2n+2)! = (2n+2)(2n+1)(2n)!$ . Now take another look at $a(n+1)/a(n)$ and see if you still get the same result.

Quote:

Originally Posted by poirot

Find the limits, if they exists, of
i) $100(ln(x))/x^3$ as n (x, not n, presumably?) tends to infinity

ii) $(sinx-x)/(cosx -x^2-1)$ as n (x again?) tends to 0

My working i) pretty certain its infinity but not sure how to show it

ii) no idea on this one.

L'Hôpital's rule should do these (apply it repeatedly until you get an answer). For i), you can guess what the answer should be (and it's not infinity) if you remember the general principle that ln(x) tends to infinity more slowly than any positive power of x.
April 12th 2011, 02:25 AM
poirot

ok I got 0.5, 0 and non exsistent respectively. Can I use l'hopital's rule for sequences or just functions? Thank for the help.

I am doing an analysis exam next month on limits of sequences, series and functions. I don't suppose you have any tips as I often feel like i'm stumbling in the darkness trying to figure out which method to use.