91Ó°ÊÓ

A geometric series is one having the form \(a+a r+a r^{2}+a r^{3} \cdots,\) where \(a, r \in \mathbb{R}\). (The first term in the sum is \(a,\) and beyond that, the \(k\) th term is \(r\) times the previous term.) Prove that if \(|r|<1,\) then the series converges to \(\frac{a}{1-r} .\) Also, if \(a \neq 0\) and \(|r| \geq 1,\) then the series diverges. (If you need guidance, you may draw inspiration from Example \(13.10,\) which concerns a geometric series with \(\left.a=r=\frac{1}{2} .\right)\)

Prove the absolute convergence test: Let \(\sum a_{k}\) be a series. If \(\sum\left|a_{k}\right|\) converges, then \(\sum a_{k}\) converges. (Your proof may use any of the above exercises.)

Prove that if a sequence diverges to infinity, then it diverges.

Prove that the constant sequence \(c, c, c, c, \ldots\) converges to \(c,\) for any \(c \in \mathbb{R}\).

Prove that \(\lim _{x \rightarrow \infty} \sin (x)\) does not exist.