Converging and diverging Series

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The discussion focuses on verifying the application of convergence tests for three series, with an emphasis on a corrected version of question 1. Participants confirm that the original work appears accurate, noting that multiple tests may apply to determine convergence. There is a query about the usefulness of calculating the first n terms as a verification method, with responses indicating that this approach may not always be effective. The conversation highlights the complexity of convergence and divergence in series analysis. Overall, the importance of using appropriate convergence tests is underscored.
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Homework Statement
Deduce whether these three series are converging or diverging
Relevant Equations
Convergence tests
Would somebody be kind enough to check whether I've picked the right convergence tests for each of these and reached the right answers? There are no solutions in the book.

Also, is there a method I can use to determine if I'm right - does calculating the first n terms help?

Thank you
Edit: meant to say it's for all integers one and over. Plus, the denominator in question 1 is meant to read 3n-2.
 

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Here's corrected question 1
 

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penroseandpaper said:
Homework Statement:: Deduce whether these three series are converging or diverging
Relevant Equations:: Convergence tests

Would somebody be kind enough to check whether I've picked the right convergence tests for each of these and reached the right answers? There are no solutions in the book.
Your work looks fine to me (including the edited version of question 1). Sometimes there are multiple convergence tests that work, so there might not be only one way to determine whether a sequence converges.
penroseandpaper said:
Also, is there a method I can use to determine if I'm right - does calculating the first n terms help?
For some sequences, calculating the first n terms doesn't help. For example, ##s_n = \{ (-1)^n\}, n \ge 1##.
penroseandpaper said:
Thank you
Edit: meant to say it's for all integers one and over. Plus, the denominator in question 1 is meant to read 3n-2.
 
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