Explanation:

To recognize the convergence of the series, we need to use the ratio test, which claims that for the series , and , if l is greater than 1, the series diverges, if together is equal to 1, the series may for sure converge, conditionally converge, or diverge, and also if l is less than 1 the series is (absolutely) convergent.

You are watching: Use the ratio test to determine whether the series is convergent or divergent.

For our series, us get

*

Using the nature of radicals and exponents come simplify, us get

*

L is better than 1 therefore the series is divergent.

 


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Example concern #113 : ratio Test


Determine whether the collection converges or diverges:

*


Possible Answers:

The collection converges


The collection may certain converge, conditionally converge, or diverge


The collection diverges


The series conditionally converges


Correct answer:

The collection converges


Explanation:

To determine the convergence of the series, we must use the ratio test, which claims that because that the series , and , if l is greater than 1, the collection diverges, if l is equal to 1, the collection may certain converge, conditionally converge, or diverge, and also if l is much less than 1 the collection is (absolutely) convergent.

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For our series, us get

*

Using the nature of radicals and exponents come simplify, we get

*

 

 


Possible Answers:

The series is divergent 


The collection is convergent, but not certain convergent. 


The collection is absolutely convergent, and therefore convergent 


The series is conditionally convergent. 


The collection is either divergent, conditionally convergent, or certain convergent. 




Explanation:

*

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The proportion test can be provided to prove the an infinite collection

*
 is convergent or divergent. In some cases, however, the proportion test might be inconclusive. 

Define:

*
 

 If...

*
 then the series is either conditionally convergent, certain convergent, or divergent.

 ______________________________________________________________

 

To compute the limit, we an initial need to compose the expression for 

*

 

*

 

*

 

Now us can uncover the limit, 

*

 

*

 

We deserve to cancel all determinants with the 

*
 exponent. 

 

*

 

Now the negative factor in the molecule

*
 will publication with 
*
 in the denominator, return this is trivial due to the fact that we space taking the absolute value regardless.