00:01
For this, problem we are to find the radius and interval of convergence of the series summation from n equals 0 to infinity of x minus 7 raise a power of n all over n raised to the fifth power plus 1.
00:13
So for this, we'll have to use the ratio test.
00:16
Now let a sub n be equal to x minus 7 raise a pair of n all over n raised to the fifth power plus 1.
00:25
Then our a sub n plus 1 this is equal to x minus 7 raise to n plus 1 over n plus 1 raise to the 5th power plus 1 so the limit as n approaches infinity of the absolute value of a sub n plus 1 over a sub n this is equal to the limit as n approaches infinity of the absolute value of x minus 7 raised n plus 1, all over n plus 1, raise to the fifth power plus 1, times the reciprocal of a sub n, that's n raised the 5th power plus 1, all over x minus 7 raised n.
01:06
So we could cancel out the x minus 7 raise to n.
01:11
You have x minus 7 here left, and that will give us absolute value of x minus 7 times the limit as n approaches infinity.
01:21
Of you have n raised to the fifth power plus one over n plus one raise to the fifth power plus one since they have the same degree then this is just one over one or one that's dividing their leading coefficients and from here we have absolute value of x minus 7 and since you want this to converge this has to be less than one that's our radius of convergence and now for the interval, if absolute value of x minus 7 is less than 1, that means negative 1 is less than x minus 7, and x minus 7 is less than positive 1.
02:06
Adding 7 both sides, we have 6 less than x, less than 8.
02:16
Now we want to check if the series converges at the end points...
