Question

Find the interval I and radius of convergence R for the given power series. (Enter your answer for interval of convergence using interval notation.)

Σ [infinity]ₖ=₁ (-1)ᵏ/8ᵏ(x-7)ᵏ

Answer 1

To find the interval of convergence and radius of convergence for the given power series, we can use the ratio test. The ratio test states that if the limit as n approaches infinity of |a(n+1) / a(n)| is less than 1, then the series converges.

Step-by-step explanation:

To find the interval of convergence and radius of convergence for the given power series, we can use the ratio test. The ratio test states that if the limit as n approaches infinity of |a(n+1) / a(n)| is less than 1, then the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive. In this case, the power series is Σ (-1)^k / 8^k * (x-7)^k. Let's apply the ratio test:

| ( (-1)^(k+1) / 8^(k+1) * (x-7)^(k+1) ) / ( (-1)^k / 8^k * (x-7)^k ) |

Simplifying, we have | (-1) / 8(x-7) | = 1 / (8|x-7|).

Using the ratio test, we take the limit as n approaches infinity:

lim(n->∞) (1 / (8|x-7|)) = 1 / (8|x-7|).

We can see that the limit is less than 1 for all x except x=7. Therefore, the series converges for |x-7| < 8, and diverges for |x-7| > 8. The interval of convergence is (-1, 15) and the radius of convergence is 8.