Limit of Sequence a_n: Find Limit as n->∞

In summary, the conversation was about finding the limit of a sequence, specifically for the function a_n = 2^n/(3^n + 1). The speaker separated the fraction into (1/3)(2/3)^n and simplified it to limit = 1/3 lim_(n->infinity) (2/3)^n. They then asked for guidance on finding the limit of (2/3)^n, to which the other speaker explained that it is a general result that for any real number a between -1 and 1, the limit of a^n is zero. They also mentioned using the geometric series and the necessary criterion for a series to converge. The conversation ended with the speaker understanding and thanking the other speaker
  • #1
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[tex]a_n = 2^n/(3^n + 1)[/tex]

Find the limit as n -> infinity

So I separated the fraction into [tex] (1/3)(2/3)^n [/tex]

Then limit = 1/3 [tex] lim_(n->infinity) (2/3)^n. [/tex]

So, my question is, how do you find the limit [tex] (2/3)^n. [/tex] Is analysis of functions simply enough, i.e. looking at how fast they increase?
 
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  • #2
How did you transform a_n into [tex] (1/3)(2/3)^n [/tex]?

Btw - is this for a real analysis class or some intro calculus class?
 
  • #3
Oops, sorry, I guess I didn't look hard enough at my post.

It's really [tex] a_n = \frac{2^n}{3^{n+1}} [/tex]

And it's Calculus II
 
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  • #4
Oh ok.

Didn't you see the general result that if -1<a<1, then [itex]a^n\rightarrow 0[/itex]?

If not, and if you've seen a little bit of series theory, you can prove it this way: You know that the geometric series [itex]\sum a^n[/itex] converges for -1<a<1 and diverges otherwise. And you've also seen the basic necessary criterion for a series to converge, namely that the general term [itex]a_n[/itex] of any converging series goes to zero in the limit n-->infty.

So you can make use of these two fact by saying "I know that the geometric series of general term [itex]a_n=a^n[/itex] (where -1<a<1) converges, hence it must be that [itex]a_n=a^n\rightarrow 0[/itex] for -1<a<1."
 
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  • #5
Ok, I get what you're saying...but why is -1<a<1 here?
 
  • #6
I proved that for any real number a btw -1 and 1, the limit of a^n is zero.

In particular, for a=2/3, we have that the limit of (2/3)^n is zero. Which is what you were wondering about.
 
  • #7
Ooh, ok, I understand now.

Thanks!
 

Related to Limit of Sequence a_n: Find Limit as n->∞

What is a limit of a sequence?

A limit of a sequence, denoted as limn→∞ an, is the value that the terms of the sequence approach as n (the index of the terms) approaches infinity. In other words, it is the value that the terms of the sequence get closer and closer to as n increases without bound.

How do you find the limit of a sequence?

To find the limit of a sequence, you can use two methods: the algebraic method and the graphical method. The algebraic method involves finding a formula for the n-th term of the sequence, then taking the limit as n approaches infinity. The graphical method involves plotting the terms of the sequence on a graph and observing the trend as n increases.

Why is finding the limit of a sequence important?

Finding the limit of a sequence is important because it helps us understand the behavior of the sequence and determine if it converges (approaches a finite value) or diverges (does not approach a finite value). It also allows us to make predictions about the long-term behavior of the sequence.

What is the difference between a finite and infinite limit of a sequence?

A finite limit of a sequence means that the terms of the sequence approach a specific finite value as n approaches infinity. This indicates that the sequence converges. On the other hand, an infinite limit of a sequence means that the terms of the sequence do not approach a specific finite value and continue to increase or decrease without bound. This indicates that the sequence diverges.

What are some common misconceptions about limits of sequences?

One common misconception is that a sequence must approach a specific value in order to have a limit. In reality, a sequence can also approach infinity or negative infinity and still have a limit. Another misconception is that a sequence must have a limit in order to be a valid sequence, when in fact some sequences may not have a limit at all.

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