Unformatted text preview: 1 < r < 0 then  r n  =  r  n and 0 <  r  < 1, so the sequence converges. In summary, { r n } converges precisely when1 < r ≤ 1 in which case lim n →∞ r n = n if1 < r < 1 1 if r = 1 Sometimes we will not be able to determine the limit of a sequence, but we need to know whether it converges. In some cases we can determine this without being able to compute the limit. A sequence is called increasing or sometimes strictly increasing if a i < a i +1 for all i . It is called nondecreasing or sometimes (unfortunately) increasing if a i ≤ a i +1 for all i . Similarly a sequence is decreasing if a i > a i +1 for all i and nonincreasing if a i ≥ a i +1 for all i . If a sequence has any of these properties it is called monotonic . EXAMPLE 10.11 The sequence ‰ 2 i1 2 i ² ∞ i =1 = 1 2 , 3 4 , 7 8 , 15 16 , . . . ,...
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 Spring '07
 JonathanRogawski
 Math, Calculus, Sequences And Series, lim, Order theory, Monotonic function

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