00:01
Hi, here we are asked to find the taylor series of a function f center at 4 if the nth derivative of that function at 4 is given by this formula.
00:14
And then we are asked to find the radius of convergence of the taylor series.
00:19
Now we know that the taylor series formula for a function f center at a point a is given by this, is the infinite sum, is a series.
00:30
We need to end the terms are like this the nth derivative of the function calculated at the point a over n factorial times x minus a to the power n here we know this and of course we need to just add these terms so we will have that the series the tailor series of this function is equal to the nth derivative of the function calculated at 4 over n factorial or maybe let me just write it split it a little bit so it's going to be easier to see so the nth derivative calculated at a times 1 over n factorial times um okay so for us a is 4 times x minus 4 to the power n so this is going to be the sum from 0 to infinity of minus 1 to the n times n factorial over 3 to the n times n plus 1 times 1 over n factorial times x minus 4 to the n.
01:45
Of course this and this cancel out.
01:48
So the series, the taylor series we are looking for is.
01:53
Minus 1 to the power n over 3 to the n times n plus 1.
02:03
Let me just create some space here.
02:11
Okay, times x minus 4 to the power n.
02:16
So this is our taylor series...