00:01
In this question, we are asked to find the center of the power series, the radius of convergence and the interval of convergence.
00:08
To find the center, we'll factor out 4.
00:13
We'll get negative 1 to the n times 4 to the n multiplied by x minus 1 half to the n.
00:27
Since our series is in the powers of x minus 1 half, that means that the center of the power series equals 1 half.
00:43
To find the radius of convergence, we'll use the ratio test.
00:51
We need to calculate the limit of the absolute value of a n plus 1 over a n as n goes to infinity, where a n is the general term of the series.
01:06
Since we are working with absolute values, we can ignore the negative sign.
01:12
To get a n plus 1, we simply need to replace n by n plus 1.
01:15
We'll get 4 to the n plus first power times x minus 1 half to the n plus first divided by the square root of n plus 4 multiplied by the reciprocal of a n.
01:42
We can cancel 4 to the n and we can cancel x minus 1 half to the n.
01:48
And when n goes to infinity, n plus 4 over square root of n plus 3 over the square root of n plus 4 goes to 1.
01:59
And in the limit, we'll get the absolute value of 4 multiplied by x minus 1 half.
02:09
And by the ratio test, the series converges if this limit is less than 1.
02:14
That means the absolute value of x minus 1 half must be less than 1 quarter.
02:21
And this means this immediately gives us the radius of convergence, which is 1 quarter.
02:31
Finally, let's find the interval of convergence.
02:33
We are almost there.
02:35
We just need to check the endpoints...