Sequence Homework Help

Question

If a sequence c₁,c₂,c₃, ... has limit K then the sequence e^c1,e^c2,e^c3, ... has limit e^K . Use this fact together with l'Hopital's rule to compute the limit of the sequence given by:

b_n = (1 + 4.3/n)ⁿ

Answer 1

If c_n = ln b_n, then c_n = n ln(1 + 4.3/n) = 4.3 ln(1 + 4.3/n)/(4.3/n). If h = 4.3/n, then h -> 0 as n -> ∞. Hence,

lim_{n -> ∞} c_n = lim_{h -> 0} 4.3 ln(1 + h)/h = 4.3 lim_{h -> 0} ln(1 + h)/h

……………= 4.3 lim_{h -> 0} 1/(1 + h) [by L’Hôpital’s rule]

…………….= 4.3

By the result you quoted, e^c_n -> e^4.3 as n -> ∞. Since b_n = e^c_n, then lim b_n = e^4.3.