- #1

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I have the following sequence

[tex]

\begin{array}{l}

a_n = ( - 1)^n \left( {\frac{n}{{n + 1}}} \right) \\

\\

\mathop {\lim }\limits_{n \to \infty } \left[ {( - 1)^n \left( {\frac{n}{{n + 1}}} \right)} \right] \\

\end{array}

[/tex]

Direct substitution yields [tex]

( - 1)^\infty \left( {\frac{\infty }{\infty }} \right)

[/tex]

I tried manipulating it into a form in which I could apply L'Hopital's Rule.

[tex]

\displaylines{

{\rm Let y} = \mathop {\lim }\limits_{n \to \infty } \left[ {( - 1)^n \left( {\frac{n}{{n + 1}}} \right)} \right] \cr

\cr

\ln y = \mathop {\lim }\limits_{n \to \infty } \ln \left[ {( - 1)^n \left( {\frac{n}{{n + 1}}} \right)} \right] \cr

\cr

= \mathop {\lim }\limits_{n \to \infty } \left[ {\ln ( - 1)^n + \ln (n) - \ln (n + 1)} \right] \cr

\cr

= \mathop {\lim }\limits_{n \to \infty } \left[ {n\ln ( - 1) + \ln (n) - \ln (n + 1)} \right] \cr

\cr

\ln ( - 1) = undefined \cr}

[/tex]

The answer is below. How did the book arrive at that answer? How did they go through and calculate the limit? Solutions manuals are so wonderfully detailed :)

http://img70.imageshack.us/img70/7812/answer5ck.jpg [Broken]

[tex]

\begin{array}{l}

a_n = ( - 1)^n \left( {\frac{n}{{n + 1}}} \right) \\

\\

\mathop {\lim }\limits_{n \to \infty } \left[ {( - 1)^n \left( {\frac{n}{{n + 1}}} \right)} \right] \\

\end{array}

[/tex]

Direct substitution yields [tex]

( - 1)^\infty \left( {\frac{\infty }{\infty }} \right)

[/tex]

I tried manipulating it into a form in which I could apply L'Hopital's Rule.

[tex]

\displaylines{

{\rm Let y} = \mathop {\lim }\limits_{n \to \infty } \left[ {( - 1)^n \left( {\frac{n}{{n + 1}}} \right)} \right] \cr

\cr

\ln y = \mathop {\lim }\limits_{n \to \infty } \ln \left[ {( - 1)^n \left( {\frac{n}{{n + 1}}} \right)} \right] \cr

\cr

= \mathop {\lim }\limits_{n \to \infty } \left[ {\ln ( - 1)^n + \ln (n) - \ln (n + 1)} \right] \cr

\cr

= \mathop {\lim }\limits_{n \to \infty } \left[ {n\ln ( - 1) + \ln (n) - \ln (n + 1)} \right] \cr

\cr

\ln ( - 1) = undefined \cr}

[/tex]

The answer is below. How did the book arrive at that answer? How did they go through and calculate the limit? Solutions manuals are so wonderfully detailed :)

http://img70.imageshack.us/img70/7812/answer5ck.jpg [Broken]

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