Determining if the sequence convergers or diverges

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In summary, the sequence \frac{tan^{-1} n}{n} converges to 0 as n approaches infinity. This is because the top of the fraction approaches a finite value (pi/2) while the bottom gets infinitely larger, resulting in a limit of 0. Therefore, the sequence converges.
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shamieh
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Determine if the sequence converges ot diverges. If it converges, find the limit

\(\displaystyle \frac{tan^{-1} n}{n}\)

So I'm thinking that I can say tan inverse is \(\displaystyle \frac{\frac{\pi}{2}}{n}\) as n--> infinity is going to be some number over infinity = 0? so therefore it converges to 0?
 
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shamieh said:
Determine if the sequence converges ot diverges. If it converges, find the limit

\(\displaystyle \frac{tan^{-1} n}{n}\)

So I'm thinking that I can say tan inverse is \(\displaystyle \frac{\frac{\pi}{2}}{n}\) as n--> infinity is going to be some number over infinity = 0? so therefore it converges to 0?

Yes, since your top goes to a finite value and your bottom gets infinitely larger, the limit of this function is 0. So yes your sequence converges to 0.
 

FAQ: Determining if the sequence convergers or diverges

What is the definition of a convergent sequence?

A convergent sequence is a sequence of numbers that approaches a single fixed value as the number of terms increases.

How can I determine if a sequence is convergent or divergent?

To determine if a sequence is convergent or divergent, you can use various methods such as the limit comparison test, the ratio test, or the root test. These tests involve evaluating the limit of the sequence and comparing it to known values to determine if the sequence approaches a fixed value or increases/decreases without bound.

What is the difference between a convergent and divergent sequence?

A convergent sequence approaches a fixed value as the number of terms increases, while a divergent sequence does not approach a fixed value and either increases or decreases without bound.

What happens if a sequence is neither convergent nor divergent?

If a sequence does not approach a fixed value and does not increase/decrease without bound, it is considered to be oscillating or alternating and is neither convergent nor divergent.

Why is it important to determine if a sequence converges or diverges?

Determining if a sequence converges or diverges is important in many areas of mathematics and science. It allows us to make predictions about the behavior of a sequence and can help us understand the underlying patterns and relationships in data. Additionally, convergence and divergence of sequences are used in various mathematical calculations and proofs.

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MHB No problem, happy to help! (Glad to hear it)
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