Question

does the power series of 3^n converge/diverge? what test would I use?

Answer 1

|an+1/an| < 1
so
3^(n+1)/3^n = 3
this is not less than 1, so the series does not converge, but diverges

Answer 2

just by imagining what happens to the series as n increases you see that \[3^n\] increases in size, \[3^n < 3^{n+1}\] due to the positive exponent, so then with some intuition to get you started you can then use a test to see if your intuition is correct

Answer 3

like beezlebub did

Answer 4

thanks to the both of you :)