```Date: Oct 3, 2013 5:42 PM
Author: quasi
Subject: Re: Sequence limit

konyberg wrote:>quasi wrote:>> quasi wrote:>> >konyberg wrote:>> >>Bart Goddard wrote:>> >>>>> >>> What is lim_{n -> oo}  |sin n|^(1/n) >> >>> >> >>> where n runs through the positive integers. >> >>> >> >>> Calculus techniques imply the answer is 1.>> >>> But the same techniques imply the answer is 1>> >>> if n is changed to x, a real variable, and that>> >>> is not the case, since sin x =0 infinitely often.>> >>> >> >>> Anyone wrestled with the subtlies of this problem?>> >>>>> >>> E.g., can you construct a subsequence n_k such>> >>>>> >>> that sin (n_k) goes to zero so fast that the>> >>> exponent can't pull it up to 1?>> >>>> >>In general a^0 = 1. lim (n goes inf) 1/n = 0. Then the value>> >>of sin(n) doesn't change that a^0 = 1.>> >>> >Your logic is flawed.>> >>> >Let f(n) = 1/(2^n).>> >>> >Then f(n)^(1/n) = 1/2 for all nonzero values of n, >> >hence the limit, as n approaches infinity, of >> >f(n)^(1/n) is 1/2, not 1.>> >> Are there infinitely many positive integers n such that >> >>    |sin(n)| < 1/(2^n)>> >> >??>> >>> >If so, then the limit of the sequence>> >>> >   |sin(n)|^(1/n), n = 1,2,3, ...>> >>> >does not exist. In particular, it would not be equal to 1.>> >>> >In fact, the original question can be recast as:>> >>> >Does there exist a real number c with 0 < c < 1 such that>> >>> >the inequality>> >>> >   |sin(n)| < c^n>> >>> >holds for infinitely many positive integers n?>>Yes I was a bit hasty here. But sin(n) is limited from -1 to +1>(your function isn't limited), Sure it is.For positive integers n, the function   f(n) = 1/(2^n)satisfies 0 < f(n) <= 1/2>and the limit of 1/n is 0. I would think that the limit is 1.I agree that in this case, the limit is probably equal to 1,But in general, if f,g are functions such that, for all positive integers n,   (1) 0 < |f(n)| <= 1   (2) 0 < g(n)   (3) g(n) --> 0 as n --> oothe question as to whether or not the limit, as n --> oo, of   f(n)^g(n)exists, and if so, to what value, cannot be answered withoutmore information about the functions f,g.In particular, for this question, it doesn't matter in theleast whether the expression 0^0 is regarded as either   undefined   equal to 1   equal to 0   equal to 1/2 (hey, split the difference)   equal to some other constant>The debate will still be what 0^0 is equal to :)Which has no relevance to the OP's question.quasi
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