00:01
For this question, we need to find the radius of convergence of the series from n equals 1 to infinity of n times x to the n.
00:09
So in order to find the radius of convergence, i need to apply the ratio test.
00:16
Now the ratio test tells us that if the limit as n approaches infinity of the absolute value of a sub n plus 1 divided by a sub n is less than 1, then the sum of all the values of a sub n converge.
00:32
So you'll notice that in the series we're given my a sub n is going to equal n times x to the n.
00:40
So we have a sub n is n x to the n.
00:43
This means that a sub n plus 1 is going to equal n plus 1 times x to the n plus 1.
00:51
Now go ahead and apply ratio test.
00:54
So i have the limit as as n approaches infinity of the absolute value of n plus 1 times x to the n plus 1 divided by nx to the n.
01:09
I'm going to simplify what's in the absolute value bars by expanding the numerator.
01:15
So i'm going to first distribute this x to the n plus 1 to both terms in the parentheses.
01:22
So this gives us the limit as n approaches infinity of the absolute value of nx to the n plus 1 plus x to the n plus 1 divided by nx to the n.
01:41
Now simplify by expanding out the exponential expressions x to the n plus 1.
01:48
You can do this using the product rule for exponents.
01:53
So this allows us to rewrite what we have as the limit as n approaches infinity of n times x to the n x to the first power...