00:01
In this question, we are asked to find the taylor series centered at 8 for the function f satisfying the given condition.
00:11
The nth derivative for the function f at x equals 8 is negative 1 to the n times n factorial over 6 to the n times n plus 4.
00:23
Recall that the taylor series formula is the series f n of c divided by n factorial multiplied by x minus c to the nth power.
00:36
So this is a general formula.
00:40
In our case, c equals to 8.
00:46
So we can rewrite this as the series f n of 8 divided by n factorial multiplied by x minus 8 to the nth power.
00:57
That's why the series is called series centered at 8 because it's in the powers of x minus 8.
01:06
Now we'll replace f n of 8 by its formula.
01:12
We'll get negative 1 to the n multiplied by n factorial divided by 6 to the n times n plus 4 times n factorial times x minus 8 to the nth power and from 0 to infinity.
01:27
We can cancel n factorial and we'll get the series negative 1 to the n multiplied by 1 over 6 to the n times n plus 4 times x minus 8 to the nth power.
01:50
This is a taylor series for the function f centered at 8.
01:58
Now we need to find, we are asked to find the radius of convergence.
02:03
To do that, we will use the ratio test...