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Show that the following sequences diverge:

  • {1+(1)n}
  • {n}
  • Solution (a) The terms of the sequence are 0,2,0,2,0,2,. See Figure 8. Since the terms alternate between 0 and 2, the terms of the sequence {1+(1)n} do not approach a single number L. So, the sequence {1+(1)n} is divergent.

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    (b) The terms of the sequence {sn}={n} are 1,2,3,4,. Given any positive number M, we choose a positive integer N>M. Then whenever n>N, we have sn=n>N>M. That is, the sequence {n} diverges to infinity. See Figure 9.