Mastering the Series Ratio Test: A Comprehensive Guide

[nameVoid]

 

[Mark44]

What's your question?

 

[nameVoid]

is it correct to apply the ratio test on An>=an and then use the results in the comparison test where bn=An for bn convergence and bn>an

 

[Mark44]

is it correct to apply the ratio test on An>=an and then use the results in the comparison test where bn=An for bn convergence and bn>an

I have no idea what you're saying here. You have An, an, and bn. How do these relate to your original series?

The first test you did was a comparison test, comparing with \sum 1/n!. When you use the comparison test, you should already know whether the series you're comparing to converges or diverges. If you are comparing \sum a_n to a convergent series, you have to show that a_n <= the corresponding term in your convergent series. If you are comparing \sum a_n to a divergent series, you have to show that a_n >= the corresponding term in the divergent series.

Also, the comparison test and the ratio test have to be used on series without negative terms. Your cosine series has negative terms.

 

[jtyler05si]

You could use the ratio test on the original problem; remember when using the ratio test you are taking the limit of the abs value of your function. You could also use the squeeze theorem along with the comparison test.