6. Let k > 0 be a constant and consider the important sequence {kn}. Its behaviour as n ® ¥ will depend on the value of k.

(i) State the behaviour of the sequence as n ® ¥ when k = 1 and when k = 0.

(ii) Prove that if k > 1 then kn ® ¥ as n ® ¥

(hint: let k = 1 + t where t > 0 and use the fact that (1 + t)n > 1 + nt.

(iii) Prove that if 0 < k < 1 then kn ® 0 as n ® ¥ .7. Given a geometric series with first term a > 0 and common ration r > 0 prove that a finite sum to infinity exists if and only if r < 1 and show that in this case the sum to infinity is .
(hint: use the results of Q6 above, and recall that the sum of the first n terms is ).Problem #6
(a) When , we have as ; when , we have
as .
(b) If , we can assume that for some , then we have
as .
Thus as .
(c) If , then , where . From (b), we know that as , then . Therefore, as .
Problem #7
The geometric series is , then the sum of the first terms isFrom the result in problem #6, if , then as . For and , converges if and only if . At this time, as . Therefore, the sum of all terms is .

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