Limits of Sequences: Manipulating Equations for Standard Limits

In summary, the conversation is about solving three limit problems involving exponents. The answers to these limits are already provided, so the person is trying to manipulate the equations into a form where they can use standard limits. They are struggling to understand how to approach the problems and are seeking help. The conversation also mentions using the formula (n+1)^k = n^k(1+1/n) to simplify the equations.
  • #1
tedwillis
13
0

Homework Statement


Have a few limits that I'm stuck on:

a) lim n->infinity (n(n+1)^(n+1))/(n+2)^(n+2))

b) lim n->infinity (n^n/(n+3)^(n+1))

c) lim n->infinity n^(-1)^n

I've tried my best to understand what to do solve these, but can't get it. We've been given answers to standard limits, so I think we need to manipulate the above equations into a form where we can use standard limits.


Homework Equations


Answers are:
a) 1/e

b) 0

c) infinity


The Attempt at a Solution

 
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  • #2
tedwillis said:
a) lim n->infinity (n(n+1)^(n+1))/(n+2)^(n+2))
(n+1)k = nk(1+1/n)k. Apply that top and bottom. Do any of your standard limits look like (1+1/n)k (where k perhaps depends on n)?
 
  • #3
There are several ways to interpret each of those sequences, largely due to them having so many exponents. You should write them up in LaTeX. Once you do that, I'll give it a go and see if I can give you a push in the right direction.

What "standard limits" do you have?
 

1. What are the limits of a sequence?

The limit of a sequence is the value that the terms of the sequence approach as the index increases without bound. It is the ultimate behavior of the sequence.

2. How do you determine the limit of a sequence?

To determine the limit of a sequence, you can examine the pattern of the terms and see if they approach a specific value as the index increases. Alternatively, you can use mathematical techniques such as the squeeze theorem or the ratio test to find the limit.

3. What happens if a sequence has no limit?

If a sequence has no limit, it means that the terms do not approach a specific value as the index increases. This could indicate that the sequence is divergent, meaning it does not have a finite limit. It could also mean that the sequence oscillates between two or more values, never approaching a single limit.

4. Can a sequence have multiple limits?

No, a sequence can only have one limit. This is because the limit is the ultimate behavior of the sequence, and if the terms approach different values, there is no single value that the sequence is approaching.

5. How do limits of sequences relate to limits of functions?

Limits of sequences and limits of functions are closely related concepts. The limit of a sequence is the same as the limit of a function at infinity. This means that as the index of the sequence increases without bound, the terms of the sequence approach the same value that the function approaches as the input increases without bound.

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