00:10
We're going to use a known tailor series to find a tailor series for this function f of x equals e to the x minus 1 over x and this series will be centered at zero.
00:36
We're going to start with the series for e to the x.
00:40
In the series for e to the x from k equals zero to infinity of one over k factorial times x to the k.
01:08
Now term by term this is one plus x squared over two factorial.
01:23
Plus x cubed over 3 factorial plus x to the 4th over 4 factorial and that pattern continues.
01:47
Now this means that when we have e to the x minus 1, it'll just be the series for e to the x but subtract the 1.
02:01
So, e to the x minus 1 is x plus x squared over 2 factorial plus x cubed over 3 factorial plus x to the 4th over 4 factorial plus and that pattern continues.
02:39
Now when we divide by x, we'll have e to the x minus 1 divided by x.
02:49
What we're doing is taking each term and dividing by x.
02:55
X divided by x will be 1.
02:57
The next term will be x over 2 factorial.
03:03
The next term will be x squared over 3 factorial.
03:11
The next term will be x cubed over four factorial, and the pattern continues.
03:24
And now when we write this series, we have the series for k equals zero to infinity.
03:35
The numerator, the coefficient in the numerator is always one, and your denominator is always a k plus one.
03:43
So, when k is zero, your denominator is one.
03:48
When k is one, you have a denominator of two, actually two factorial.
03:55
When k is three, you'd have three plus one, which is the four factorial.
04:03
So no matter what k is, see notice that in this term your k is zero.
04:10
Then in this term, your k is 1.
04:16
In this term, your k is 2, and in this term your k is 3.
04:23
So that denominator is always one more than what k is, or the factorial.
04:29
The number is one more than what k is, and then we take the factorial of that number.
04:34
Now then we have x to the k.
04:40
So here's our series.
04:45
Now, we were also asked to find the radius of convergence...
