Finding the Limit of a Convergent Sequence

In summary, the conversation discusses how to determine if a given sequence converges or diverges, and how to find the limit if it does converge. The sequence in question is provided as a link, and the conversation mentions using L'Hopital's rule to find the limit, but this approach becomes complicated. The conversation also mentions considering the effect of (-1)^{n+1} on the sequence.
  • #1
StrangeCharm
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12

Homework Statement


Determine whether the sequence converges or diverges. If it converges, find the limit.
Here's the sequence: http://www4a.wolframalpha.com/Calculate/MSP/MSP89541ea2ag9dg617bcd6000050d52e94i67ei593?MSPStoreType=image/gif&s=39&w=66.&h=44.

Homework Equations


N/A

The Attempt at a Solution


I know that the sequence is convergent if the limit exists; however, I'm having difficulty finding the limit. I tried finding the limit of the sequence as n-->infinity using L'Hopital's rule, but that got messy. I'm not sure what to do, especially because of all the terms.
 
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  • #2
StrangeCharm said:
I tried finding the limit of the sequence as n-->infinity using L'Hopital's rule, but that got messy..

Find [itex] lim_{n \rightarrow \infty} \frac{n}{n + \sqrt{n}} [/itex] and then think about the effect of [itex] (-1)^{n+1} [/itex].
 
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Related to Finding the Limit of a Convergent Sequence

What is a convergent sequence?

A convergent sequence is a sequence of numbers that approaches a specific value, called the limit, as the number of terms in the sequence increases. In other words, the terms in the sequence get closer and closer to the limit value as the sequence continues.

Why is finding the limit of a convergent sequence important?

Finding the limit of a convergent sequence is important because it allows us to understand the behavior of the sequence and make predictions about its future terms. It also helps us to solve problems in various fields of mathematics, such as calculus and analysis.

How do you find the limit of a convergent sequence?

To find the limit of a convergent sequence, we need to observe the pattern of the sequence and determine if it is approaching a specific value. If the sequence is approaching a limit, we can use various methods such as the squeeze theorem, the ratio test, or the root test to find the limit value.

What happens if a convergent sequence does not have a limit?

If a convergent sequence does not have a limit, it is called a divergent sequence. This means that the terms in the sequence do not approach a specific value, and the sequence does not have a predictable pattern. Divergent sequences can have several types of behavior, such as growing without bound or oscillating between multiple values.

Can a sequence have multiple limits?

No, a sequence can only have one limit. If a sequence has multiple limits, it is not a convergent sequence. In other words, if the terms in a sequence are approaching more than one value, the sequence is divergent and does not have a limit.

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