How Does the Limit of the Sequence sqrt(n)*(sqrt(n+1)-sqrt(n)) Approach 1/2?

In summary: This is done by showing that for any given epsilon, there exists N such that for all n>N, the absolute value of the difference between the limit and the sequence is less than epsilon.
  • #1
tarheelborn
123
0

Homework Statement



Evaluate lim sqrt(n)*[sqrt(n+1)-sqrt(n)]



Homework Equations



sqrt(n)/[sqrt(n+1)+sqrt(n)] = 1/sqrt[1+(1/n)]+1

The Attempt at a Solution



I know that limit sqrt(n) = Infinity and that limit (sqrt(n+1)-sqrt(n)) = 0. And I know that sqrt(n)*(sqrt(n+1)-sqrt(n)) = sqrt(n)/(sqrt(n+1)+sqrt(n)).

I believe the limit of the product of these sequences is 1/2, but I am not sure how to get there. I need to do an epsilon proof of the limit and I am not sure how to solve the equation in 2. in terms of epsilon. Thanks for your help.
 
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  • #2
You can't find a limit using epsilon/delta.

Evaluating the limit is almost immediate after simple algebra [tex](a+b)(a-b)=a^2-b^2[/tex], and yes the limit is 1/2.

If you still want to prove it by definition you should for any given [tex]\epsilon>0\ find\ N>0\ so\ \forall\ n>N\ |a_n-1/2|<\epsilon [/tex]
 
Last edited:
  • #3
tarheelborn said:

Homework Statement



Evaluate lim sqrt(n)*[sqrt(n+1)-sqrt(n)]



Homework Equations



sqrt(n)/[sqrt(n+1)+sqrt(n)] = 1/sqrt[1+(1/n)]+1

The Attempt at a Solution



I know that limit sqrt(n) = Infinity and that limit (sqrt(n+1)-sqrt(n)) = 0. And I know that sqrt(n)*(sqrt(n+1)-sqrt(n)) = sqrt(n)/(sqrt(n+1)+sqrt(n)).

I believe the limit of the product of these sequences is 1/2, but I am not sure how to get there. I need to do an epsilon proof of the limit and I am not sure how to solve the equation in 2. in terms of epsilon. Thanks for your help.
What do you mean "how to solve the equation in 2 in terms of epsilon"? You can get that equation using estro's hint and use it to show that the limit of the sequence is indeed equal to 1/2 as n goes to infinity.
 

FAQ: How Does the Limit of the Sequence sqrt(n)*(sqrt(n+1)-sqrt(n)) Approach 1/2?

What is the definition of a limit of a product of sequences?

The limit of a product of sequences is the value that the product of the two sequences approaches as the number of terms in each sequence approaches infinity. It represents the maximum or minimum possible value of the product.

How is the limit of a product of sequences calculated?

The limit of a product of sequences can be calculated by finding the limits of each sequence individually, and then multiplying them together. It is important to note that the limits of both sequences must exist for the limit of the product to exist.

What are the properties of limits of products of sequences?

Some important properties of limits of products of sequences include the product rule, which states that the limit of a product is equal to the product of the limits of the individual sequences, and the zero product property, which states that if one of the sequences has a limit of 0, then the product of the two sequences will also have a limit of 0.

How is the limit of a product of sequences used in real-world applications?

The concept of limits of products of sequences has many real-world applications, such as in economics, physics, and engineering. For example, it can be used to predict the maximum or minimum value of a product, such as the maximum profit a company can make with a certain production rate, or the minimum amount of material needed to construct a bridge.

What are some common misconceptions about limits of products of sequences?

One common misconception is that the limit of a product of sequences is always equal to the product of the limits of the individual sequences. This is only true if both sequences have finite limits and the product rule applies. Another misconception is that the limit of a product of sequences always exists, when in fact, there are cases where the limit does not exist or is infinite.

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