infinite series

[lotus776]

I'm having trouble determining which way to test the convergence/divergence of a series that has the same degree in both numberator and denominator.
For Example: n/(3n+1), I know it is divergent but how do I provide proof (which test)?

thanks

[gumpzilla]

Well, what's the limit of each term in the infinite series as n -> infinity?

[lotus776]

1/3...this proves divergence based on the test for divergence. Is that how you would have done it?

[gumpzilla]

Rather than apply the names of some formulas, let's just think about what it means that that limit is 1/3. When you get far enough out, each new term that you add is going to contribute roughly 1/3 to the value of your sum. Given that, is it possible for the series to converge?

[lotus776]

I don't understand. I don't see why it's not possible, I'm not doubting you but I don't get it. If the limit of the series->1/3 as n->infinity than what's the prob? please explain

[lotus776]

no, because it's not approaching any particular number but continues on until infinity, thus divergent. Interesting. I don't understand what you mean "each new term that you add is going to contribute roughly 1/3 to the value of your sum" I don't understand this, but I do understand conceptually what you are saying.

[gumpzilla]

Conceptually is the best way to put it, because I don't think you would find it particularly illuminating for me to put it technically. The idea basically is this: as n gets huge, each additional term in the series is 1/3. So if X is the sum of the first 1000000 terms, X + 1 will be very close to the sum of the first 1000003 terms. Notice that no matter how many terms I pick, something like this will be true. So there's no limit to how big the sum gets, and so it doesn't converge.

[lotus776]

thank you, I understand. excellent explanation