Question

Determine whether the sequence converges or diverges. if it converges, find the limit.
a sub n= (7+7n)/(9+3n)

Answer 1

just take the limit as \(n\to\infty\)

Answer 2

I really dont know how to approach these types of problems. I am trying to learn just from a book. And as you may know, calculus is hard to learn from a book

Answer 3

we want to know if the sequence approaches a finite value as n goes to infinity, so just take the limit
if it exists, it means that that is what the sequence converges to

Answer 4

it converges.....and limit is 7/3

Answer 5

L'Hopital's rule might help

Answer 6

so you take the derivative so it would be 7/3 and thats how you get the answer

Answer 7

If you use L'Hopital's rule, you will take the derivative of the top and bottom separately.

Answer 8

l'Hospital is unnecessary; you can just divide top and bottom by n

Answer 9

why do you divide by n?

Answer 10

Or just observe.
Neglect the terms which dont approach infinity.
Take ratio of the coefficient of n.
Although it's the same thing, really.

Answer 11

\[\lim_{n\to\infty}{7+7n\over9+3n}\]if we try to take the limit now we get\[\frac\infty\infty\]but if we divide top and bottom by n first we get\[\lim_{n\to\infty}{\frac7n+7\over\frac9n+3}={\frac7\infty+7\over\frac9\infty+3}={0+7\over0+3}=\frac73\]

Answer 12

do you always approach infinity?

Answer 13

yes because that is what they are asking by asking if \(\{a_n\}\) converges
they are asking of \(\{a_n\}\) approaches some finite value as \(n\to\infty\)
that is, do the terms of the sequence ever "settle down" to any real number

Answer 14

oh ok thank you!