- #1
opticaltempest
- 135
- 0
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
[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
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