I've solved most! just need with 9, 10, 14.

would love if i could check my answers to 7, 11, 16 for which i got

7. radius of convergence = 1

11. interval of convergence = [1, 3)

16. converges for abs value {x} < 1. not sure how to put that into interval of the numbers back in?

7. Correct

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9. By the ratio test, the series will converge if



The limit reduces to



where because is given. So the series converges when



This means the radius of convergence is .

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10. By the ratio test, the series converges if



The limit is



and so the radius of convergence is 1.

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11. Incorrect. By the root test, the series converges for



which means the series converges for all , and so the interval of convergence is .

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For 14 and 16, it'll probably be too late to edit this post by the time you see this. You can try posting the remaining problems in a new question.