00:01
To figure out the radius of convergence, we'll first just look at where the series converges.
00:05
So we'll use the ratio test.
00:07
Take limit as n goes to infinity of absolute value of a .n plus 1 over a .n.
00:12
Where a .n is this whole thing here, including the x values.
00:16
So that's limit as n goes to infinity of absolute value of x minus 2 to the n plus 1, divided by n plus 1 to the power of n plus 1, dividing by a .n, which is, same thing as multiplying by the reciprocal.
00:39
Okay, so x minus 2 to the n plus 1 divided by x minus 2 to the n.
00:44
That just simplifies to x minus 2.
00:51
Okay, that's one of the nice things about having a common base.
00:56
And here we don't quite have a common exponent, but n plus 1 to the power of n plus 1 can be written as n plus 1 to the n times n plus 1.
01:06
So if we do that, then we can rewrite this thing like this...