In the following series x is a real number. In each case, use the ratio test to determine the radius of convergence of the series. Analyze the behavior at the endpoints in order to determine the interval of convergence.

a. Summation n=0 to infinity (n(x^n))/(n^2 + 2)

b. Summation n=1 to infinity [(n^2)(x-2)^n]/(2^n)

c. Summation n=1 to infinity [(4^n)(x^n)]/(n^2)

To determine the radius of convergence for each series, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.

a. For the series: ∑(n=0 to infinity) [(n(x^n))/(n^2 + 2)]
Let's apply the ratio test:

lim(n→∞) |[(n+1)(x^(n+1))/(n^2+2)] * [(n^2+2)/(n(x^n))]|

Simplifying the expression, we get:

lim(n→∞) |(x(n+1))/(n+1)*(n^2+2)/(n^2)|

As n approaches infinity, the terms with n in the expression become negligible, and we are left with:

lim(n→∞) |x/(n+1)|

Since the limit is independent of x and equals 0, the ratio test is inconclusive. To analyze the behavior at the endpoints and determine the interval of convergence, we need to check the convergence at x = -R and x = R.

When x = -R, the series becomes:

∑(n=0 to infinity) [(n(-R)^n)/(n^2+2)]

Applying the ratio test, we can simplify it as previously shown and find that the limit is 0. Therefore, the series converges at x = -R.

When x = R, the series becomes:

∑(n=0 to infinity) [(n(R)^n)/(n^2+2)]

Again, applying the ratio test and simplifying, we find that the limit is also 0. Therefore, the series converges at x = R.

Since the series converges for both x = -R and x = R, the interval of convergence for this series is (-R, R).

b. For the series: ∑(n=1 to infinity) [(n^2)(x-2)^n]/(2^n)
Let's apply the ratio test:

lim(n→∞) |[(n+1)^2(x-2)^(n+1)]/(2^(n+1))] * [(2^n)/[(n^2)(x-2)^n]]|

Simplifying the expression, we get:

lim(n→∞) |(x-2)(n+1)^2/(2n^2)|

As n approaches infinity, the terms with n in the expression become negligible, and we are left with:

lim(n→∞) |(x-2)/(2)|

Since the limit is independent of n and equals |(x-2)/2|, the ratio test tells us that the series converges when |(x-2)/2| < 1. Simplifying this inequality, we get |x-2| < 2, which implies -2 < x-2 < 2. Solving for x, we find that the interval of convergence is (-4, 6).

c. For the series: ∑(n=1 to infinity) [(4^n)(x^n)]/(n^2)
Let's apply the ratio test:

lim(n→∞) |[(4^(n+1))(x^(n+1))]/[(n+1)^2]*[(n^2)/[(4^n)(x^n)]]|

Simplifying the expression, we get:

lim(n→∞) |(4x(n+1)^2)/(n+1)^2|

As n approaches infinity, the terms with n in the expression become negligible, and we are left with:

lim(n→∞) |4x|

Since the limit |4x| is independent of n, the ratio test tells us that the series converges when |4x| < 1. Simplifying this inequality, we get -1/4 < x < 1/4. Therefore, the interval of convergence for this series is (-1/4, 1/4).

In summary:
a. The radius of convergence is inconclusive, and the interval of convergence is (-R, R).
b. The radius of convergence is 2, and the interval of convergence is (-4, 6).
c. The radius of convergence is 1/4, and the interval of convergence is (-1/4, 1/4).