00:01
In this question, we are asked to determine the radius of convergence of the series.
00:04
And we will use the ratio test.
00:11
By the ratio test, we need to calculate the limit of the absolute value of a n plus one over a n as n goes to infinity, where a n is the general term of the series.
00:29
To get a n plus one, just replace n by n plus one.
00:32
We'll get the limit of absolute value of x minus three to the n plus first power divided by x minus three to the n.
00:46
We can cancel x minus three to the n, and the remaining expression doesn't depend on n, so we'll get the absolute value of x minus three in the limit.
00:56
And by the ratio test, this must be less than one for the series to converge.
01:02
That means that the radius of convergence of the series n over two to the n multiplied by x to the n, n from zero to infinity.
01:26
We will again use the ratio test.
01:35
We will calculate the limit of the absolute value of a n plus one over a n...