Question

Find the radius of convergence of the power series. (If you need to use or –, enter INFINITY or –INFINITY, respectively.) [infinity] (−1)n xn 8n n = 0

Answer 1

To find the radius of convergence of the power series, we can use the ratio test. The radius of convergence for the given power series is x < -1 or x > -1.

Step-by-step explanation:

To find the radius of convergence of the power series, we can use the ratio test. The ratio test states that if the absolute value of the ratio of consecutive terms approaches a limit less than 1 as n approaches infinity, then the series converges. Let's apply this test to the given power series:

Using the ratio test:

Calculate the absolute value of the ratio of consecutive terms: |(8(n+1))/n * x|

Take the limit as n approaches infinity: lim(n->∞) |(8(n+1))/n * x|

Determine the value of x that makes the limit less than 1: |(8(n+1))/n * x| < 1

Express the inequality in terms of x: |8(x+1)/x|< 1

Simplifying the above inequality, we get:

8|x+1|< |x|

Using the properties of absolute value, we can rewrite the inequality as two separate cases:

Case 1: x+1 > 0

Case 2: x+1 < 0

Solving each case separately, we find that the radius of convergence is x < -1 for Case 1, and x > -1 for Case 2.

Therefore, the radius of convergence for the given power series is x < -1 or x > -1.