Help with sum of infinite series using the root test.

In summary: Thus, our limit is:\lim_{n\to\infty} \left( \frac{n}{n+1} \right)^n = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^{-n} = e^{-1} = \frac{1}{e}.In summary, the given series converges and the limit of the function is 1/e. The root test can be used to show convergence, and the limit can be evaluated using the properties of logarithms and the definition of the constant e.
  • #1
Homework Statement

Does it converge or diverge?

Sum n=0 to infinity : (n/(n+1))^(n^2)

The attempt at a solution

I know I need to use the root test.
But what I get is ...

Limit to infinity : (n/(n+1))^n

It seems that the n/(n+1) would go to 1 because when you multiply the top and bottom by n^-1 to get 1/(1+1/n) the 1/n goes to zero and you have 1/1. 1^n is 1, and according to the root test, it is inconclusive. However, the series converges and apparently the limit goes to 1/e, but I'm not sure how.

Also, does the latex syntax not work in preview mode or something?
 
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  • #2
To evaluate

[tex]\lim_{n\rightarrow\infty} (1+1/n)^n[/tex]

try finding the limit of the log of the function.
 
  • #3
vela said:
To evaluate

[tex]\lim_{n\rightarrow\infty} (1+1/n)^n[/tex]

try finding the limit of the log of the function.

I have no idea how.
 
  • #4
Are you saying that you do not know how to find a logarithm??

If [itex]y= (1+ 1/n)^n[/itex], then [itex]log(y)= n log(1+ 1/n)[/itex]. What is the limit of that as n goes to infinity? Perhaps L'Hopital's rule or writing log(1+ 1/n) as a Taylor's series will help.
 
  • #5
iluvphysics said:
Homework Statement

Does it converge or diverge?

Sum n=0 to infinity : (n/(n+1))^(n^2)

The attempt at a solution

I know I need to use the root test.
But what I get is ...

Limit to infinity : (n/(n+1))^n

It seems that the n/(n+1) would go to 1 because when you multiply the top and bottom by n^-1 to get 1/(1+1/n) the 1/n goes to zero and you have 1/1. 1^n is 1, and according to the root test, it is inconclusive. However, the series converges and apparently the limit goes to 1/e, but I'm not sure how.

Also, does the latex syntax not work in preview mode or something?

[tex]\lim_{n\to\infty} \left( \frac{n}{n+1} \right)^n = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^{-n} = ?[/tex]

Recall:
[tex]\lim_{n\to\infty} \left( 1 + \frac{a}{n^k} \right)^{bn^k} = e^{ab}[/tex]
 

1. What is the root test and how does it work?

The root test is a method used to determine the convergence or divergence of an infinite series. It involves taking the nth root of the absolute value of the terms in the series and then evaluating the limit as n approaches infinity. If the resulting limit is less than 1, the series converges. If it is greater than 1, the series diverges. If it is equal to 1, the test is inconclusive.

2. When should I use the root test?

The root test is most useful for determining the convergence or divergence of series with terms that involve exponents or radicals. It is also helpful when the terms in the series do not alternate in signs.

3. How do I apply the root test to a specific series?

To apply the root test to a series, you will need to take the nth root of the absolute value of each term. Then, you will evaluate the limit as n approaches infinity. If the resulting limit is less than 1, the series converges. If it is greater than 1, the series diverges. If it is equal to 1, the test is inconclusive.

4. Can the root test be used to prove convergence or divergence?

Yes, the root test can be used to prove convergence or divergence of a series. If the resulting limit is less than 1, it proves that the series converges. If it is greater than 1, it proves that the series diverges. If it is equal to 1, further tests may be needed to determine the convergence or divergence of the series.

5. Are there any other methods for determining the convergence or divergence of infinite series?

Yes, there are several other methods, such as the ratio test, integral test, and comparison test. It is important to use multiple tests and techniques to confirm the convergence or divergence of a series.

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