The geometric series
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6. Let k > 0 be a constant and consider the important sequence {kn}. It's 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 ).
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Solution Summary
The geometric series is clearly evaluated.
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Problem #6
(a) When , we have as ; when , we have
as .
(b) If , we can assume that for some , then ...
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