Determine whether the following series converges

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Comparison Test.In summary, using the Ratio Test, we can determine that the series converges by simplifying the expression and using the Comparison Test.
  • #1
Chadlee88
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Could someone please explain how they got from step 1 to step 2, i don't know what hap'd to the factorial part :S

Question: Determine whether the following series converges.

n= infinity
Sum ln (n)
n=1 n!

Using the Ratio Test: lim absolute ((An+1)/(An))
n->infinity

Step 1. => lim (ln(n+1)/(n+1)!) x (n!/(ln (n))

Step 2. => lim ln(n+1)/((n+1)ln(n))


Link to solution: http://www.maths.uq.edu.au/courses/MATH1051/Semester2/Tutorials/prob9sol.pdf

It's question 1. vi)
 
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  • #2
n!/(n + 1)! = 1/(n + 1).

If you write out the factorials for n = 1, 2, 3 you should see why this is the case.
 
  • #3
Using this equality e^x >= x + 1 > x for all x, we have e^n > n for all n. Thus, ln(n) < n for all n. This implies 0 < ln(n)/n! < 1/(n-1)! for all n>1
The series (Sum 1/(n-1)!) is convergent, so is the series (Sum ln(n)/n!)
 

Related to Determine whether the following series converges

1. What is the definition of convergence of a series?

The convergence of a series is the property that determines whether the sum of the terms in the series approaches a finite limit as the number of terms increases to infinity.

2. How do you determine if a series converges or diverges?

To determine if a series converges or diverges, one can use different tests such as the comparison test, the ratio test, or the integral test. These tests involve analyzing the behavior of the terms in the series to determine if they approach a finite limit or if they oscillate or grow without bound.

3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series that converges regardless of the order in which the terms are added. Conditional convergence refers to a series that only converges when the terms are added in a specific order. In other words, rearranging the terms of an absolutely convergent series will not change the sum, but rearranging the terms of a conditionally convergent series can result in a different sum or even divergence.

4. Can a series converge to a value other than the sum of its terms?

No, a series can only converge to the sum of its terms. This is a fundamental property of series convergence known as the limit of a sequence theorem.

5. Are there any shortcuts or tricks to determine convergence?

While there may be some patterns or behaviors that can make it easier to determine the convergence of a series, there are no shortcuts or tricks that work for all series. It is important to understand and apply the various convergence tests to determine the convergence or divergence of a series.

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