Convergence of a sequence

In summary, the conversation discusses finding the convergence or divergence of a sequence involving nth roots and the use of various tests and techniques such as the root test, log rules, and l'Hopital's rule. Ultimately, it is determined that the sequence converges to 5. The conversation then shifts to a new question, but it is suggested that a new thread be created for it.
  • #1
grothem
23
1

Homework Statement


Determine if the sequence is convergent or divergent
{[tex]\sqrt[n]{3^n+5^n}[/tex]}


Homework Equations





The Attempt at a Solution


I know I need to take the limit to find if it converges or diverges. But I'm not really sure what I need to do to it to take the limit.
 
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  • #2
Take the log of the sequence. Write 3^n+5^n=5^n(1+(3/5)^n). Does that help?
 
  • #3
no need to that even, use the root test, and from there you can tell
 
  • #4
I thought you could only apply the root test to a series, not a sequence, or does it not matter?
 
  • #5
grothem said:
I thought you could only apply the root test to a series, not a sequence, or does it not matter?

Right. The root test is not helpful for sequences.
 
  • #6
so after taking the log I came up with 5(1+(3/5)) = 8. Which I don't think is the right answer, did I do something wrong?
 
  • #7
grothem said:
so after taking the log I came up with 5(1+(3/5)) = 8. Which I don't think is the right answer, did I do something wrong?

Yes, quite a bit. Review your rules of logs. Try again. log([5^n*(1+3^n/5^2)]^(1/n)). What's your first step?
 
  • #8
oh ok...

so I bring the 1/n in front of the ln(5^n*(3/5)^n)

now can I just take the limit from there?
 
  • #9
Careful, you dropped a very important '1'. Now use log(a*b)=log(a)+log(b). Then the power rule again.
 
  • #10
I came up with Limit as n tends to infinity of (ln(3)*(3/8)^n + ln(5)*(5/8)^n)
and with those being geometric sequences with r < 1, the sequence is convergent.
 
  • #11
grothem said:
I came up with Limit as n tends to infinity of (ln(3)*(3/8)^n + ln(5)*(5/8)^n)
and with those being geometric sequences with r < 1, the sequence is convergent.

That's a long way from being correct. How did you get that?
 
  • #12
I took the natural log and I got:

Lim 1/n * ln(3^n+5^n)

Then applying L'hospitals:

(3^n*ln(3)+5^n*ln(5))/(3^n+5^n)

then I simplified from there.
 
  • #13
grothem said:
I took the natural log and I got:

Lim 1/n * ln(3^n+5^n)

Then applying L'hospitals:

(3^n*ln(3)+5^n*ln(5))/(3^n+5^n)

then I simplified from there.

If you want to go that way then fine. Then the 'simplification' went wrong. Try dividing numerator and denominator by 5^n. What's the limit of 3^n/5^n?
 
  • #14
Dick said:
If you want to go that way then fine. Then the 'simplification' went wrong. Try dividing numerator and denominator by 5^n. What's the limit of 3^n/5^n?

ok. The limit of 3^n/5^n = 0

So after dividing through by 5^n, I'm left with ln(5). Which would be a divergent series
 
  • #15
grothem said:
ok. The limit of 3^n/5^n = 0

So after dividing through by 5^n, I'm left with ln(5). Which would be a divergent series

It's a divergent series, but it's a convergent sequence. It converges to ln(5). But now remember you took the log of the original sequence.
 
  • #16
Dick said:
It's a divergent series, but it's a convergent sequence. It converges to ln(5). But now remember you took the log of the original sequence.

Oh ok. So raising ln(5) to power e gives me 5. So sequence converges to 5.
Thanks a lot for the help!
 
  • #17
You're welcome. I think it would be a good exercise to try and do this without l'Hopital. You don't really need it, just use the rules of logs.
 
  • #18
how can i proof that ([sin0.4n][/npi])2 converges and ([sin0.4n][/npi]) diverges? n is between -infinity and +infinity
 
  • #19
What is the sequence? It isn't clear in the way you've written down.
 
  • #20
jlu said:
how can i proof that ([sin0.4n][/npi])2 converges and ([sin0.4n][/npi]) diverges? n is between -infinity and +infinity

Open a new thread for a new question. Don't tag it onto an old thread. It won't get the attention it deserves. And try to post the question more legibly on the new thread. What is ([sin0.4n][/npi])2? Do you mean (sin(theta*4n)/(n*pi))^2??
 

What is the definition of convergence of a sequence?

Convergence of a sequence is a mathematical concept that refers to a sequence of numbers approaching a specific value or limit as the number of terms in the sequence increases. This limit is known as the limit of the sequence, and if the sequence converges, it means that the terms get closer and closer to this limit.

How can you determine if a sequence converges?

To determine if a sequence converges, you can use the definition of convergence which states that for a sequence to converge, the terms must get arbitrarily close to the limit of the sequence as the number of terms in the sequence increases. This can also be checked by using convergence tests, such as the ratio test or the root test.

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

A convergent sequence is a sequence that approaches a specific limit as the number of terms increases. In contrast, a divergent sequence is a sequence that does not have a limit or approaches infinity as the number of terms increases. In simpler terms, a convergent sequence has a finite value, while a divergent sequence does not.

What is the importance of convergence in mathematics?

Convergence is an essential concept in mathematics as it helps us understand the behavior of sequences and series. It allows us to determine if a sequence or series will have a finite value or will continue to increase without bound. Convergence is also crucial in many practical applications, such as in physics and engineering, where it is used to model and predict real-world phenomena.

What are some real-life examples of convergence?

Convergence can be observed in many real-life situations, such as compound interest in finance, where the amount of money in an account approaches a specific value as the number of compounding periods increases. It can also be seen in the growth of populations, where the population size approaches a stable value as resources become limited. Other examples include the movement of a pendulum, the decay of radioactive substances, and the speed of a falling object as it approaches terminal velocity.

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