Answer
(a) $1,2,1,4,1,6,...$
(b) ${a_n} = \left\{ {\begin{array}{*{20}{c}}
{n,}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{odd}}}\\
{\dfrac{1}{{{2^n}}},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{even}}}
\end{array}} \right.$, ${\ \ \ \ \ \ \ }$ for $n = 1,2,3,...$
(c) ${a_n} = \left\{ {\begin{array}{*{20}{c}}
{\dfrac{1}{n},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{odd}}}\\
{\dfrac{1}{{n + 1}},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{even}}}
\end{array}} \right.$, ${\ \ \ \ \ \ \ }$ for $n = 1,2,3,...$
(d)
The sequence in part (a) diverges.
The sequence in part (b) diverges.
The sequence in part (c) converges. It converges to zero.
Work Step by Step
(a) Starting with $n=1$,
$1,2,1,4,1,6,...$
(b) We have
$1,\dfrac{1}{{{2^2}}},3,\dfrac{1}{{{2^4}}},5,\dfrac{1}{{{2^6}}},...$
Thus, the general term is
${a_n} = \left\{ {\begin{array}{*{20}{c}}
{n,}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{odd}}}\\
{\dfrac{1}{{{2^n}}},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{even}}}
\end{array}} \right.$, ${\ \ \ \ \ \ \ }$ for $n = 1,2,3,...$
(c) We have
$1,\dfrac{1}{3},\dfrac{1}{3},\dfrac{1}{5},\dfrac{1}{5},\dfrac{1}{7},\dfrac{1}{7},\dfrac{1}{9},\dfrac{1}{9},...$
Thus, the general term is
${a_n} = \left\{ {\begin{array}{*{20}{c}}
{\dfrac{1}{n},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{odd}}}\\
{\dfrac{1}{{n + 1}},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{even}}}
\end{array}} \right.$, ${\ \ \ \ \ \ \ }$ for $n = 1,2,3,...$
(d)
Part (a): The sequences $\left\{ {{a_n}} \right\}$, where
${a_n} = \left\{ {\begin{array}{*{20}{c}}
{1,}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{odd}}}\\
{n,}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{even}}}
\end{array}} \right.$, ${\ \ \ \ \ \ \ }$ for $n = 1,2,3,...$
does not converge to some finite limit as $n \to \infty $, but oscillate between $1$ and $\infty $. Therefore, by definition it diverges.
Part (b): The sequences $\left\{ {{a_n}} \right\}$, where
${a_n} = \left\{ {\begin{array}{*{20}{c}}
{n,}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{odd}}}\\
{\dfrac{1}{{{2^n}}},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{even}}}
\end{array}} \right.$, ${\ \ \ \ \ \ \ }$ for $n = 1,2,3,...$
does not converge to some finite limit as $n \to \infty $, but oscillate between $\infty $ and $0$. Therefore, by definition it diverges.
Part (c): The sequences $\left\{ {{a_n}} \right\}$, where
${a_n} = \left\{ {\begin{array}{*{20}{c}}
{\dfrac{1}{n},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{odd}}}\\
{\dfrac{1}{{n + 1}},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{even}}}
\end{array}} \right.$, ${\ \ \ \ \ \ \ }$ for $n = 1,2,3,...$
approaches zero for both odd and even terms as $n \to \infty $. Thus, it converges to zero.
$\mathop {\lim }\limits_{n \to \infty } {a_n} = 0$
