00:01
Find the radius of convergence of the following power series.
00:07
So we have the sum minus 1 to the power of n times n cubed times x to the power of n divided by 4 to the power of n for n ranging between 1 to infinity.
00:27
So we start by finding the radius of convergence from the result of the ratio test.
00:33
Test.
00:44
So according to the ratio test, if the limit when n approaches infinity of a n plus 1 over a n, defined as l, is less than 1, then the series converges.
01:14
So let's evaluate this quantity l, as it's going to be related to our radius of convergence.
01:24
So in our case, we have l equal to the limit when n approaches infinity of the absolute value of minus 1 to the power of n plus 1 times n plus 1 cubed times x to the power of n plus 1 divided by 4 to the power of n plus 1.
01:54
And we're going to multiply this by 1 over a n.
01:56
That is 4 to the power of n divided by minus 1 to the power of n times n cubed times x to the power of n.
02:10
So our negative powers are going to simplify.
02:16
Not negative powers, i mean minus 1 to the power of n.
02:19
We're going to simplify it due to the absolute value.
02:20
Value, so we're going to have the limit with n, first infinity, of the opposite value of n plus 1 to the power of 3, divided by n cubed.
02:39
Now, x to the power of n is going to simplify with this power, and for the power of n, it's going to simplify with this power.
02:47
So we will be left with x over 4.
03:03
As x over 4 dot the pen, we can remove from our limit, so we obtain the absolute value of x over 4 times the limit when n approaches infinity of n plus 1 cube over n cube...