5.6.1

[STARTS AT 10:06]

Instructor: Our last ratio test question, hopefully getting to be old hat at this, so just dive in. We have our ratio. And that is going to be n plus 1 cubed over 3 to the n plus 1 multiplied by, again, taking the reciprocal of this for our term, 3 to the n. And we have n cubed. 

Again, all the terms are positive here. So the absolute values are useless. They don't actually serve a real purpose. But they do-- they weren't necessary in example 53, if you noticed that. So some things simplify. These two become a 3 in the denominator, 3, and then we have our n plus 1 cubed and an n cubed. 

And for that same reasoning in the last one, this ratio, or this limit, because of our ratio, is going to be 1/3, which is less than 1. And so by the ratio test, n cubed over 3 to the n converges.