Convergence: Epsilon-N Definition

[UOAMCBURGER]

Homework Statement

Homework Equations

epsilon - N definition of convergence [/B]

The Attempt at a Solution

Not sure how to determine whether this sequence converges or not. Thought it could have something to do with the fact that e^2 and e^3 are just constants, so when n > infinity those e terms become negligible in comparison. If you had to apply convergence definition how could you do that without L? [/B]

 

[SammyS]

Homework Statement

View attachment 232999

Homework Equations

epsilon - N definition of convergence

The Attempt at a Solution

Not sure how to determine whether this sequence converges or not. Thought it could have something to do with the fact that e^2 and e^3 are just constants, so when n > infinity those e terms become negligible in comparison. If you had to apply convergence definition how could you do that without L?

Do you know rules for logarithms ?

Such as $\ \ln(a\cdot b) =\ln(a)+\ln(b)\,?$

What is $\ \ln(e^2) \, ?$

 

[UOAMCBURGER]

Do you know rules for logarithms ?

Such as $\ \ln(a\cdot b) =\ln(a)+\ln(b)\,?$

What is $\ \ln(e^2) \, ?$

oh yes i do. Update: I used L'Hospital's Rule to find that the sequence converges to 2. But now separating the logarithms you get 2+ln(n^2)/3+ln(n), so would that change my answer I get using L'Hospital's rule? or can i assume a_n converges to 2 and then use the definition with L = 2?

 

[SammyS]

oh yes i do. Update: I used L'Hospital's Rule to find that the sequence converges to 2. But now separating the logarithms you get 2+ln(n^2)/3+ln(n), so would that change my answer I get using L'Hospital's rule? or can i assume a_n converges to 2 and then use the definition with L = 2?

Did you mean $\ \displaystyle \frac{2+\ln(n^2)}{3+\ln(n) } \,?$

If you write a "fraction" all on one line, you need to enclose the numerator and denominator each in parentheses; as in (2+ln(n^2))/(3+ln(n)) .

Also, you may further find that it helps to review rules for logarithms in general.

Another helpful one here is $\ \displaystyle \ln(a^M) = M\ln(a)\ .$

In addition to this:
You know that the limit is 2, so take the difference between 2 and $\ \displaystyle \frac{2+\ln(n^2)}{3+\ln(n) } \,.\$ That should be useful for an $\epsilon - N\$ argument. It will also give you what you need to write $\ \displaystyle \frac{2+\ln(n^2)}{3+\ln(n) } \$ in a simplified form.