Show that the following sequences diverge:

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.