Given f(n) = $\displaystyle \frac{1}{4}n^2(n+1)^2$. Evaluate $\displaystyle \liminf\frac{f(n)}{f(2n)}$.
*show working and explanation PLS
Hey mate,
f(n) = (1/4)n^2(n+1)^2
Thus,
f(2n) = (1/4)(2n)^2(2n+1)^2 = n^2(2n+1)^2
Hence,
f(n)/f(2n) = (1/4)n^2(n+1)^2/ ( n^2 (2n+1)^2) = (1/4)(n+1)^2/(2n+1)^2
to evaluate the limit of this expression as n tends towards infinity can be evaluated a number of ways, however given your title has the word differentation I am assuming your looking to use L'hopitals Rule,
lim (n-->inf) g(n)/h(n) = lim(n-->inf) g'(n)/h'(n) if g(inf)/h(inf) = inf/inf
Thus,
lim (n-->inf) (1/4)(n+1)^2/(2n+1)^2 = (1/4)lim(n-->inf) ( 2(n+1)/(4(2n+1) )
= (1/8)lim(n-->inf)( (n+1)/(2n+1) ) which once again uses inf/inf, thus employ L'hopitals again,
=(1/8)lim(n-->inf) ((1)/(2)) = (1/8)lim(n-->inf)(1/2) = (1/8)(1/2) = (1/16)
Hope this helps,
David