Converging and diverging Series

[penroseandpaper]

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.

 

[penroseandpaper]

Here's corrected question 1

 

[Mark44]

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.

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, $s_n = \{ (-1)^n\}, n \ge 1$.

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.