Using the Test for Divergence

1. \(\sum\limits_{k\,=\,1}^{\infty }87\) diverges, since \(\lim\limits_{n\,\rightarrow \,\infty }87=87\neq 0\).
2. \(\sum\limits_{k\,=\,1}^{\infty }k\) diverges, since \( \lim\limits_{n\,\rightarrow \,\infty }n=\infty \neq 0\).
3. \(\sum\limits_{k\,=\,1}^{\infty }(-1)^{k}\) diverges, since \( \lim\limits_{n\,\rightarrow \,\infty }\,(-1)^{n}\) does not exist.
4. \(\sum\limits_{k\,=\,1}^{\infty }2^{k}\) diverges, since \( \lim\limits_{n\,\rightarrow \,\infty }\,2^{n}=\infty \neq 0\).