00:01
Okay, so we want to evaluate this series, and the hint tells us to find a power series equal to this.
00:07
So the first thing to notice is that if we write 1 plus 1 minus x, then this is the geometric series.
00:14
N equals 0 to infinity x to the n, and this converges for the absolute value of x less than 1.
00:21
So if we have 1 over 1 plus x, this is the same as n equals 0 to infinity of minus 1 to the n, x to the n.
00:31
Because x is the same as minus minus x.
00:36
And the next thing to notice is that if we take log of 1 plus x and take the derivative, d by dx, of log of 1 plus x, then we end up with 1 over 1 plus x, which is this series, n equals 0 to infinity minus 1 to the n, x to the n.
00:58
So if we want the power series for the log, we just need to integrate this series.
01:02
And we can do this turn by term because this converges absolutely for mod x less than one...