00:01
In this problem, we want to find the radius and interval of convergence of the given power series.
00:06
So here we have the sum of minus 2 to the power of n times x minus 3 to the power of n divided by n squared times 5 to the power of n, for n ranging between 1 to infinity.
00:27
To determine our radius of convergence, we first want to conduct the ratio test.
00:35
According to the ratio test, we want to evaluate the quantity l, which corresponds to the limit when n approaches infinity of the absolute value of a n plus 1 over a n, and our series will converge as long as l is less than 1.
00:53
This is the criteria.
00:56
And with this criteria, we can determine the radius and interval of convergence.
00:59
So let's evaluate l.
01:01
In our case, we have that a n is equal to minus 2 to the power of n times x minus 3 to the power of n divided by n squared times 5 to the power of n.
01:22
Now we're here to determine a n plus 1.
01:24
A n plus 1 would give us minus 2 to the power of n plus 1 times x minus 3 to the power of n plus 1 divided by n plus 1 squared times 5 to the power of n plus 1.
01:43
Let's evaluate l, the limit when n approaches infinity of the absolute value of minus 2 to the power of n plus 1 times x minus 3 to the power of n plus 1 divided by n plus 1 squared times 5 to the power of n plus 1.
02:10
And this will be multiplied by the reciprocal, or the 1 over a n.
02:15
That is n squared times 5n divided by minus 2 to the power of n times x minus 3 to the power of n.
02:25
So from here we can make numerous simplifications.
02:28
Notably, our minus terms will cancel out due to the absolute value...