00:01
In this question, we are asked to determine the interval of convergence of the given series and then determine the function to which this series converges within the interval of convergence.
00:13
Alright, this series looks like a geometric series.
00:24
And the common ratio of the geometric series equals to 8 to the x minus 9.
00:31
Now recall that for a geometric series to converge, so the series, u to the n, and from 0 to infinity converges if the absolute value of u is strictly less than 1.
00:54
In our case, u equals to 8 to the x minus 9.
01:02
So r and u same in our case.
01:09
So to simplify a notation let's replace this by r here.
01:16
So this is r and the series r to then converges if the absolute value of r is less than 1.
01:25
In our case, r equals to 8 to the x minus 9, so the series converges for 8 to the x minus 9 less than 1.
01:36
This is equivalent to 8 to the x minus 9 being less than 1 and greater than negative 1.
01:46
Now we can add 9 to both side of the inequality to get that the series converges for 8 to the x less than 10 and greater than 8.
02:09
Alright, now we need to solve this inequality for x.
02:14
The left inequality is easy because it's 8 less than 8 to the x if x is greater than 1...