Showing a Sequence Diverges

Show that the following sequences diverge:

1. \(\{1+(-1)^{n}\}\)
2. \(\{n\} \)

Solution (a) The terms of the sequence are \(0, 2, 0, 2, 0, 2, \ldots \,\). 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 \(\{ s_{n}\}=\{n\} \) are \(1,\,2,\,3,\,4,\ldots.\) Given any positive number \(M\), we choose a positive integer \(N>M.\) Then whenever \(n>N,\) we have \(s_{n}=n>N>M.\) That is, the sequence \(\{n\} \) diverges to infinity. See Figure 9.