Given power series:


∑ (2x+5)^n / n4^n
n = 1

- determine the following.

A. Find the center point for the series.
B. Find the radius of convergence.
C. Determine the interval of convergence

To find the center point, radius of convergence, and interval of convergence for the given power series, we can make use of the ratio test.

A. To determine the center point, we need to find the value of x for which the power series is centered. In general, the power series of the form ∑ (aₙ(x - c)ⁿ) is centered at x = c. In this case, the given power series is (∑ (2x+5)^n / n4ⁿ), so it is centered at x = -5/2. Therefore, the center point for this series is x = -5/2.

B. The radius of convergence, denoted by R, is the distance between the center point and the nearest point at which the series diverges. The ratio test states that for a power series ∑(aₙ(x - c)ⁿ), if the limit of the absolute value of the ratio of successive terms |aₙ₊₁ / aₙ| as n approaches infinity exists, then the series converges if the limit is less than 1 and diverges if the limit is greater than 1.

In this case, we have aₙ = (2x+5)^n / n4ⁿ. Applying the ratio test, we calculate:
limit as n approaches infinity of |aₙ₊₁ / aₙ| = limit as n approaches infinity of |(2x+5)^(n+1) / (n+1)4^(n+1) * n4ⁿ|

Simplifying, we get:
limit as n approaches infinity of |aₙ₊₁ / aₙ| = limit as n approaches infinity of |(2x+5)^(n+1) / (n+1)4n+1 * n4ⁿ|

We can rewrite this expression as a product of three limits:
limit as n approaches infinity of |2x+5| * [limit as n approaches infinity of ((n⁺¹) / n) * (4ⁿ⁺¹ / 4ⁿ) * (4ⁿ / (n+1)4ⁿ)]

Simplifying further, we obtain:
limit as n approaches infinity of |2x+5| * [1 * (4 / 4) * (1 / (1 + 1/n))]

As n approaches infinity, (1 + 1/n) approaches 1, so our expression becomes:
limit as n approaches infinity of |2x+5| * 1 * 1 = |2x+5|

For the given power series to converge, the absolute value of (2x+5) must be less than 1. Therefore, the radius of convergence is R = 1.

C. The interval of convergence is the interval centered at the center point (-5/2) within which the power series converges. To determine the interval of convergence, we need to find the values of x for which the absolute value of (2x+5) is less than 1.

|2x+5| < 1
-1 < 2x+5 < 1
-6 < 2x < -4
-3 < x < -2

Therefore, the interval of convergence is (-3, -2), excluding the endpoints.