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EXAMPLE 13Determining Whether a Sequence Is Bounded from Above or Bounded from Below

  1. (a) The sequence {sn}={3nn+2} is bounded both from above and below because 3nn+2=31+2n<3 and 3nn+2>0 for all n1 See Figure 12(a).

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  2. (b) The sequence {an}={4n3} is bounded from below because 4n3>1 for all n1. It is not bounded from above because lim See Figure 12(b).
  3. (c) The sequence \{b_{n}\} =\left\{ (-1)^{n+1}n\right\} is neither bounded from above nor bounded from below.If n is odd, \lim\limits_{n\rightarrow \infty }b_{n} = \lim\limits_{n\rightarrow \infty } n =\infty, and if n is even, \lim\limits_{n\rightarrow \infty }b_{n}= \lim\limits_{n\rightarrow \infty } (-n) = -\infty. See Figure 12(c).