Which Convergence Test to Use for ∑(k(k+2))/(k+3)^2 in Calculus?

In summary, a convergence test is a mathematical method used to determine the convergence or divergence of a series. Choosing the right test is important as different tests work better for different types of series. Some common tests include the ratio, root, comparison, and integral tests. Deciding which test to use often involves trial and error and considering the properties of the series. While there are no strict rules, general guidelines include using specific tests for series involving exponents, factorials, and polynomials, and considering the behavior of the series at its endpoints.
  • #1
mickellowery
69
0

Homework Statement



∑(k(k+2))/(k+3)^2 I honestly have no idea where to start with this I was going to try a ratio test but I wasn't sure if it would be (k+1)(k+3)/(k+4)2*(k+3)2/(k(k+2)

Homework Equations





3. The Attempt at a Solution
 
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  • #2
What is the limit as k goes to infinity of

[tex]\frac{k(k+2)}{(k+3)^{2}}[/tex] ?

What does this tell you?
 
  • #3
Big Hint: Going off of what Dunkle said, use Divergence Test.
 
  • #4
Yea I was trying my hardest to make this problem way harder than it needed to be. Thanks a lot for your help.
 

FAQ: Which Convergence Test to Use for ∑(k(k+2))/(k+3)^2 in Calculus?

What is a convergence test?

A convergence test is a mathematical method used to determine whether a series (a sequence of numbers) is convergent or divergent.

Why is it important to choose the right convergence test?

Choosing the right convergence test is important because different tests are better suited for different types of series. Using the wrong test can lead to incorrect conclusions about the convergence or divergence of a series.

What are some common convergence tests?

Some common convergence tests include the ratio test, the root test, the comparison test, and the integral test.

How do I decide which convergence test to use?

Choosing a convergence test often involves trial and error. You may need to try multiple tests before finding one that works for a particular series. It can also be helpful to consider the properties of the series, such as whether it is alternating or contains factorials.

Are there any general rules for choosing a convergence test?

While there are no strict rules for choosing a convergence test, some general guidelines include using the ratio and root tests for series involving exponents, using the comparison test for series involving factorials, and using the integral test for series involving polynomials. It is also important to consider the behavior of the series at the endpoints of its interval of convergence.

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