Another sequence convergence proof

In summary, the problem asks to show that the sequence y_n, defined as the difference between the square root of n+1 and the square root of n, converges. The solution involves manipulating the terms to show that the difference is less than any given epsilon, and using a specific manipulation involving the conjugate of the denominator to simplify the expression.
  • #1
antiemptyv
34
0

Homework Statement



Let [tex]y_n := \sqrt{n+1} - \sqrt{n}[/tex] for [tex]n \in \mathbb{N}[/tex]. Show that [tex] (y_n)[/tex] converges.

Homework Equations



The Attempt at a Solution



I see that it converges to 0. I just need a nudge in the right direction at getting into [tex]| \sqrt{n+1} - \sqrt{n} - 0 | = | \sqrt{n+1} - \sqrt{n} |[/tex] to show it's less than any [tex]\epsilon > 0[/tex]. Any manipulating I've tried so far makes the terms way too big to work with.
 
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  • #2
How about

[tex]\left(\sqrt{n+1} - \sqrt{n}\right) \cdot \frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}}[/tex]
 
  • #3
ohhhh, i see it now.
 
  • #4
What do you do after
1/(sqrt{n+1)+sqrt{n}) ??
 

FAQ: Another sequence convergence proof

1. What is a sequence convergence proof?

A sequence convergence proof is a mathematical proof that shows a sequence of numbers approaches a specific value or limit as the number of terms in the sequence increases. It is used to determine if a sequence is convergent or divergent.

2. How do you prove a sequence is convergent?

To prove a sequence is convergent, you must show that as the number of terms in the sequence increases, the terms get closer and closer to a specific value or limit. This can be done by using mathematical techniques such as the epsilon-delta method or the squeeze theorem.

3. What is the difference between a convergent and a divergent sequence?

A convergent sequence is one in which the terms approach a specific value or limit as the number of terms increases. In a divergent sequence, the terms do not approach a specific value or limit, but instead, they either increase or decrease without bound.

4. Can a sequence have more than one limit?

No, a sequence can only have one limit. If a sequence has more than one limit, it is considered to be divergent.

5. What is the importance of proving sequence convergence?

Proving sequence convergence is important because it allows us to determine the behavior of a sequence as the number of terms increases. This information can be used to make predictions and solve problems in various fields such as physics, engineering, and economics.

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