Determine the limit of the convergent sequence

  • Thread starter ArthurRead
  • Start date
  • #1
16
0

Homework Statement


Determine the limit of the convergent sequence:
##a_n## =##"3/n" ^ "1/n"##
http://www.wolframalpha.com/input/?i=lim+as+x+approaches+infinity++%283%2Fx%29^%281%2Fx%29


Homework Equations



--

3. The Attempt at a Solution [/b

So I tried to get this series to look like 0/0 or ∞/∞ so I could use the L'Hopital rule.
I took the Ln of both sides. I got (ln(3/n))/n. As n approaches ∞, the numerator is -∞ and the denominator is ∞. Can
I use the L'Hopital's rule now?


THANKS!
 
Last edited:

Answers and Replies

  • #2
STEMucator
Homework Helper
2,075
140
Given the way you've presented the sequence, this rule will be extremely useful to you :

(a/b)^c = a^c / b^c

Big hint : Remember that 1 over something really really big... aka ∞ tends to 0.

Example :

lim x->∞ 1/x = 0
 
  • #3
16
0
Thank you,
So I get (3^(1/n)) / (n^(1/n))
If n approaches ∞, the the numerator is 1 because the exponent approaches 0.
So would it be sufficient for me to apply the same mentality to denominator and say
that it approaches 1? That would give me 1 for the limit of the sequence which is the correct
answer.
 
  • #4
STEMucator
Homework Helper
2,075
140
Thank you,
So I get (3^(1/n)) / (n^(1/n))
If n approaches ∞, the the numerator is 1 because the exponent approaches 0.
So would it be sufficient for me to apply the same mentality to denominator and say
that it approaches 1? That would give me 1 for the limit of the sequence which is the correct
answer.
Yup :) good job. Sometimes the most basic rules can save you the most headaches.
 
  • #5
I like Serena
Homework Helper
6,577
176

Homework Statement


Determine the limit of the convergent sequence:
a(sub n)=(3/n)^(1/n)
http://www.wolframalpha.com/input/?i=lim+as+x+approaches+infinity++%283%2Fx%29^%281%2Fx%29


Homework Equations



--

3. The Attempt at a Solution [/b

So I tried to get this series to look like 0/0 or ∞/∞ so I could use the L'Hopital rule.
I took the Ln of both sides. I got (ln(3/n))/n. As n approaches ∞, the numerator is -∞ and the denominator is ∞.


Welcome to PF, ArthurRead! :smile:

Can I use the L'Hopital's rule now?


THANKS!
Yes...
 
  • #6
I like Serena
Homework Helper
6,577
176
And no, you cannot just say that ##n^{1/n}## approaches 1.

Consider that ##n^{1/2}## approaches infinity.
And that ##2^{1/n}## approaches 1.
##n^{1/n}## could approach anything from 1 up (although it does approach 1 ;)).

To illustrate this: ##(1+{1 \over n})^n## approaches the mathematical constant e.
 
  • #7
16
0
I am sorry "I Like Serena", I don't understand what you mean by "n1/n could approach anything from 1 up". What ever value n has, you will be taking it to the power of 0. So it has to approach 1. I know I am wrong but I don't understand how.
 
  • #8
I like Serena
Homework Helper
6,577
176
You might also say that whatever value (1/n) has, you will be taking infinity to the power of this number.
So then it would have to approach infinity.
 
  • #9
16
0
That makes sense. So am I better off going with my original approach of using L'Hops rule after taking the natural log of both sides?
 
  • #10
I like Serena
Homework Helper
6,577
176
Yes.
 
  • #11
16
0
I am not getting the right answer. (ln(3/n))/n: the derivative of the numerator is (-1/n) and the derivative of the denominator is 1. Which gives me lim = 0. Did I differentiate correctly?
 
  • #12
I like Serena
Homework Helper
6,577
176
Actually, that looks good. :)

Remember which limit you approached?
 
  • #13
16
0
Ah, that was the limit of natural log of the sequence. Since ln(a sub n) = 0, (a sub n) = e^0=1. Thank you so much for guiding me through this. I feel so much better.

by the way, is there an easier way to write (a sub n)?...I'm new to this.
 
  • #14
I like Serena
Homework Helper
6,577
176
Good! :smile:

Try:
Code:
##a_n##
That comes out as ##a_n##.

Similarly
Code:
$$\lim_{n \to \infty} a_n$$
comes out as:
$$\lim_{n \to \infty} a_n$$
 
  • Like
Likes 1 person
  • #15
16
0
Got it, Thanks again
 

Related Threads on Determine the limit of the convergent sequence

Replies
9
Views
3K
Replies
4
Views
1K
  • Last Post
Replies
4
Views
6K
Replies
6
Views
4K
Replies
1
Views
403
  • Last Post
Replies
2
Views
527
  • Last Post
Replies
1
Views
952
  • Last Post
Replies
1
Views
582
Top