How would you find the radius of convergence when you end up with something like |8x - 1| < 1? Do I divide everything by 8 and get a radius of 1/8?
|8x-1| < 1
-1 < 8x-1 < 1
0 < 8x < 2
0 < x < 1/4
Would the radius be 1/4 then?
I look at it this way:
|8x - 1| < 1
8x-1 < 1 AND -(8x - 1) < 1
8x < 2 AND -8x < 0
x < 1/4 AND x > 0
0 < x < 1/4 , the same as Steve's answer.
To find the radius of convergence for a power series, we need to determine the values of x for which the series converges.
In your case, you have the inequality |8x - 1| < 1. To simplify this, we can rewrite it as -1 < 8x - 1 < 1.
Next, let's solve the two inequalities separately. First, let's solve 8x - 1 > -1:
8x - 1 > -1
8x > 0
x > 0
Now, let's solve 8x - 1 < 1:
8x - 1 < 1
8x < 2
x < 2/8
x < 1/4
Therefore, the values of x for which the series converges are 0 < x < 1/4.
To find the radius of convergence, we take the smaller value between the distance from the center of the power series to the leftmost point of convergence (0) and the distance from the center to the rightmost point of convergence (1/4).
The radius of convergence is the smaller of these two values, which in this case is 1/4.
Therefore, the radius of convergence for your power series is 1/4.