00:01
Okay, we're interested in this infinite series.
00:05
So let's try the ratio test.
00:09
So i'm gonna take the limit n goes to infinity of six to the n plus one over two to the n plus one minus one divided by six, that should be six to the n.
00:22
I'll correct that in a moment.
00:28
So i can, because this is six to the n over six to the n plus six to the n plus one over six to the n, i can factor out a six to the n and just get six.
00:41
And then i flip over the other fraction and i get two to the n minus one over two to the n plus one minus one.
00:54
Like i said, that's six to the n, not six n.
00:57
There we go, okay.
00:59
So the six comes outside of the limit, but i can't forget about it because it's part of taking the limit was that six.
01:15
Then i'll divide numerator and denominator by two to the n, okay.
01:27
And i'll take the limit of that and it's just a half.
01:34
So it's six times a half, which is three.
01:37
That's greater than one.
01:39
And so by the ratio test, it diverges.
01:52
Part b, another way to see part a just intuitively is it's six to the n is always gonna be bigger than two to the n minus one...