Im trying to prove that it approaches infinity. I am not sure how to prove the series approaches infinity.

Basically the answer says that since 4/3>1 , the series approaches infinity... but i dont know what test they are using or their rationale


Edit:

By using the Geometric Test, it could be stated that since the common ratio is greater than 1, the series will approach infinity because it just grows larger and larger

^ I think this statement would suffice in proving that the series approaches infinity. (Since it is constantly growing exponentially)

This post was edited by CamelFinger on Dec 11 2014 10:04pm

you dont need a "test", and that question has nothing to do with divergence. it's a geometric series with a common ratio greater than 1, therefore it approaches infinity. expand it out if it doesnt make sense to you.

ar^0 + ar^1 + ar^2 + ... + ar^n

ar^(n+1) > ar^(n), so you can see each term is getting larger, so the sum must be getting larger, so it approaches infinity.

please define the test you're referring to. i'm only aware of the divergence test, which has nothing to do with this. the only conclusion the divergence test makes is that it diverges or converges.

Basically this is the books answer (of which i need to explain)

Psubn = (4^n)/(3^n-1) --------------------------- To clarify n-1 <-- that is all an exponent

= 4(4/3)^n-1 -------------------------------- To clarify n-1<--- that is all an exponent

Since 4/3 is > 1, Psubn approaches infinity when n approaches infinity


I am trying to explain why the series approaches infinity

So yeah, I dont need to use a Test. If its a Geometric Series where common ratio is > 1 ..... then that should be sufficeint to prove the series goes to infinity. Correct?

This post was edited by CamelFinger on Dec 11 2014 10:19pm

it's a geometric sequence with common ratio greater than 1. that's all you really gotta say. if anything, expand it out like i did above. not sure what you're looking for. it has nothing to do with the divergence test.

What has been said is correct, since n is a positive integer and and we have a number, say x raised to the n power, we have
X^n And it is clear to see since x>1 as n increases as does our sequence exponentially thus as n increases our sequence approaches infinity.