Chapter 9.1, Problem 39ES

(a) Starting with $n=1$ write out the first six terms of the sequence $\left\{a_n\right\},$ where

$a_n={\{}_{n,\text{if }n\text{ is even}}^{1,\text{if }n\text{is}\text{odd}}$

(b) Starting with $n=1$ and considering the even and odd terms separately, find a formula for the general term of the sequence

$1,\frac{1}{2^2},3,\frac{1}{2^4},5,\frac{1}{2^6},\mathrm{...}$

(c) Starting with $n=1$ and considering the even and odd terms separately, find a formula for the general term of the sequence $1,\frac{1}{3},\frac{1}{3},\frac{1}{5},\frac{1}{5},\frac{1}{7},\frac{1}{7},\frac{1}{9},\frac{1}{9},\mathrm{...}$

(d) Determine whether the sequences in part (a), (b), and (c), converge. For those that do, find the limit.