What is a convergent sequence and a divergent sequence?

20 10 POINTS! Divergent or convergent sequence Free points!!?

• (a) Determine whether the sequence a(subscript n)= (n!/(2^n)) , n > or = 1 is convergent or divergent. if it converges find its limit. (b) Let a(subscript 1) = 1 and a(subscript n) = (sqrt(2+a(subscript n) - 1)) for n > or = 2 Prove that the sequence (a(subscript n)) converges. find its limit.

• Answer:

  I'll do the first one for you. a) We see that: a_(n + 1) = (n + 1)!/2^(n + 1) a_n = n!/2^n Then: [a_(n + 1)]/(a_n) = [(n + 1)!/2^(n + 1)]/(n!/2^n) = (n + 1)/2. Then, taking lim (n-->infinity) [a_(n + 1)]/(a_n) gives that the limit is infinity. So the series diverges. I hope this helps!