Sequence Limit Problem - # 2

[Jason76]

Does the following sequence converge or diverge. and to what value?

\(\displaystyle a_{n} = \dfrac{1 - 2n}{1 + 2n} \)

\(\displaystyle \lim n \rightarrow \infty [\dfrac{1 - 2n}{1 + 2n} ] \)

\(\displaystyle \lim n \rightarrow \infty [\dfrac{1 - 2(\infty)}{1 + 2(\infty)} ] = \) indeterminate

\(\displaystyle \lim n \rightarrow \infty [\dfrac{1 - 2}{1 + 2} ] = \dfrac{-1}{3} \) The book answer is \(\displaystyle -1\)

 

[Jason76]

I see the error now, in doing L hopital's wrong.

Does the following sequence converge or diverge. and to what value?

\(\displaystyle a_{n} = \dfrac{1 - 2n}{1 + 2n} \)

\(\displaystyle \lim n \rightarrow \infty [\dfrac{1 - 2n}{1 + 2n} ] \)

\(\displaystyle \lim n \rightarrow \infty [\dfrac{1 - 2(\infty)}{1 + 2(\infty)} ] = \) indeterminate

\(\displaystyle \lim n \rightarrow \infty [\dfrac{-2}{2} ] = -1 \)

 

[HallsofIvy]

More easily- and obviously- divide both numerator and denominator by n: \(\displaystyle \frac{\frac{1}{n}- 2}{\frac{1}{n}+ 2}\).