Convergence Tests for Sin(2n)/n^2 and (-3)^n/n!: Ratio Test vs Comparison Test

  • Thread starter badtwistoffate
  • Start date
In summary, the first question is about determining whether the series Sin(2n)/n^2 is absolutely convergent. The person asking considered using the ratio test but found it to be complicated. They suggest using the comparison test as a possible easier alternative. The second question is about the series (-3)^n/n! and the person asking used the ratio test to determine its convergence, finding it to be absolutely convergent. They provide their calculation using the ratio test and ask for confirmation if it is correct. They also mention considering the Ratio Test or d'Alambert's ratio for this series.
  • #1
badtwistoffate
81
0
help determining where the series here is abs. convergent.

its Sin(2n)/n^2, i thought about the ratio test but it gets nasty, is there a easier way?
nm, i think it is convergence if I use the comparison test? Sound right?what about (-3)^n/n!, i used the ratio test, and got 0, which means it abs. convergent since 0<1, but i don't think i did it right .
I did:
Lim n-> infinity : (-3^(n+1)/(n+1)!)(n!/-3^n)= -3 lim n!/n+1!...
does that look right?
 
Last edited:
Physics news on Phys.org
  • #2
On the 1st one: what about the comparison test to its absolute value series?

On the 2nd one: Consider the Ratio Test or d'Alambert's ratio.
 
Last edited:

1. What is the purpose of convergence tests for series?

Convergence tests for series are used to determine whether an infinite series converges or diverges. This is important in mathematics and science, as it allows us to make accurate predictions and calculations based on the behavior of a series.

2. What is the Ratio Test and how does it work?

The Ratio Test is a convergence test that compares the ratio of the absolute value of successive terms in a series. If the limit of this ratio is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is exactly 1, the test is inconclusive and another test should be used.

3. How does the Comparison Test differ from the Ratio Test?

The Comparison Test compares the given series to another known series with known convergence properties. If the known series converges and the given series is less than or equal to it, then the given series also converges. If the given series is greater than or equal to the known series and the known series diverges, then the given series also diverges.

4. Can both the Ratio Test and Comparison Test be used for any type of series?

No, the Ratio Test and Comparison Test can only be used for series with positive terms. If a series has negative terms, the Alternating Series Test or Absolute Convergence Test should be used instead.

5. How can I determine which convergence test to use for a specific series?

There is no one definitive method for determining which convergence test to use. It often involves trial and error, as well as understanding the properties and behavior of different types of series. It is also important to check for any specific conditions or exceptions for each test. Consulting with a math or science teacher/professor or using online resources can also be helpful in determining the appropriate test to use.

Similar threads

Interval of convergence of power series from ratio test
    • Calculus and Beyond Homework Help
Replies
2
Views
188
How to show that these sequences are summable?
    • Calculus and Beyond Homework Help
Replies
1
Views
259
On the ratio test for power series
    • Calculus and Beyond Homework Help
Replies
2
Views
711
Prove that this series converges uniformly on R
    • Calculus and Beyond Homework Help
Replies
4
Views
310
A problem with the convergence of a series
    • Calculus and Beyond Homework Help
Replies
5
Views
991
On pointwise convergence of Fourier series
    • Calculus and Beyond Homework Help
Replies
3
Views
417
Does the Alternating Series Test show convergence for this series?
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
The convergence Criteria ratio
    • Calculus and Beyond Homework Help
Replies
2
Views
776
Understanding the Ratio Test for Series and Its Applications
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
Infinite Series (The Ratio Test)
    • Calculus and Beyond Homework Help
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top