00:01
In this problem, we are given this limit as x goes to 0, x minus arctangent x over x cubed.
00:15
And we are going to evaluate this limit using series.
00:21
Okay, so we have this single x term here.
00:24
We don't need to do anything on it.
00:26
We have this single x cubed term.
00:29
So again, there is no need to perform a series expansion.
00:33
So the only tricky part is this arctangent function.
00:37
So now let us find a series expansion for the arctangent function.
00:45
The easiest way to do that is to consider the expression 1 over 1 plus x squared.
00:52
Because we know that arctangent is the integral of this function.
00:58
And we know how to express this guy using a series.
01:02
So we have 1 over 1 minus minus x squared dx and now we remember that 1 over 1 minus y is equal to summation from n equal to 0 to infinity y to the power n.
01:22
So we have integral summation from n equal to 0 to infinity minus x squared to the power n times dx.
01:34
Now let us put the summation outside...