00:01
Hi, from the question given that we need to determine the radius of convergence and the interval of convergence for the following power series.
00:10
So, first one is sum of n is equal to 1 to infinity minus 2 power n divided by n x to the power of n.
00:20
So, here a n is equal to minus 2 power n divided by n x to the power of n and a n plus 1 is equal to minus 2 power n plus 1 divided by n plus 1 x to the power of n plus 1.
00:35
By using the ratio test and limit n tends to infinity a n plus 1 divided by a n.
00:44
So, which is equal to limit n tends to infinity a n plus 1 is minus 2 to the power of n plus 1 divided by n plus 1 multiplied with x to the power of n plus 1.
00:59
Now, denominator of a n became the reciprocal when it is multiplied with a n plus 1.
01:07
So, multiplied with a n plus 1 divide n divided by minus 2 to the power of n x to the power of n.
01:21
So, for the simplification limit n tends to infinity cancel out the common terms.
01:30
So, we have minus 2 to the power of 1 x to the power of 1 n by n plus 1.
01:42
So, which is equal to limit n tends to infinity 2 x n by n times of 1 plus 1 by n...