More Convergence & Divergence with sequences

[shamieh]

Determine whether the sequence converges or diverges, if it converges fidn the limit.

\(\displaystyle a_n = n \sin(1/n)\)

so Can I just do this:

\(\displaystyle n * \sin(1/n)\) is indeterminate form

so i can use lopitals

so:

\(\displaystyle 1 * \cos(1/x) = 1 * 1 = 1\)

converges to 1?

 

[chisigma]

Determine whether the sequence converges or diverges, if it converges fidn the limit.

\(\displaystyle a_n = n \sin(1/n)\)

so Can I just do this:

\(\displaystyle n * \sin(1/n)\) is indeterminate form

so i can use lopitals

so:

\(\displaystyle 1 * \cos(1/x) = 1 * 1 = 1\)

converges to 1?

Your result is correct... however, if You want to use continuous functions, it is better to set $\displaystyle x=\frac{1}{n}$ and the limit becomes $\displaystyle \lim_{x \rightarrow 0} \frac{\sin x}{x}=1$. This is a 'fundamental limit' and the l'Hopital rule shouldn't be used...

Kind regards

$\chi$ $\sigma$