f(x)=x^2/4+x^3

find the power series representation...so far i found ...Óx^2n+2/4^n...is dat correct?...nd then it is asking to find the radius and interval of convergence...i found the radius to be x^2/4...so i set the radius between -1 and 1...nd then solve for x...i got x to be -2 and 2...so when i plug in -2 and 2 i cant tell if they converge or diverge? how do i do dat?

I don't understand the question
f(x)=x^2/4 +x^3
is already a power series, with only two terms. it converges everywhere.

I apologize for the confusion in my previous response. You are correct that the given function, f(x) = x^2/4 + x^3, is already a polynomial with two terms and therefore can be considered as a power series. Since a power series is an infinite series, we can expand this expression further.

To find the power series representation of f(x), we can express it as a sum of terms involving powers of x. Let's expand f(x) as a power series around the point x = 0 (also known as the Maclaurin series):

f(x) = (x^2/4) + x^3
= x^2/4 + x^3(1 + 0 + 0 + ...)

The coefficient of the nth term is given by the derivative of the function evaluated at x = 0, divided by n! (n factorial). Let's calculate the derivatives:

f'(x) = 2x/4 + 3x^2
f''(x) = 2/4 + 6x
f'''(x) = 6

Now, let's evaluate the derivatives at x = 0:

f'(0) = 0/4 = 0
f''(0) = 2/4 = 1/2
f'''(0) = 6

Substituting the derivatives into the power series representation, we get:

f(x) = x^2/4 + (1/2)x^3 + 0x^4/4! + 0x^5/5! + ...

So, the power series representation of f(x) is:

f(x) = (x^2/4) + (x^3/2) + O(x^4)
= (1/4)x^2 + (1/2)x^3 + O(x^4)

Regarding the radius and interval of convergence, we can determine them by utilizing the ratio test. The radius of convergence (R) is given by the limit:

R = lim(n->∞) |a(n)/a(n+1)|

In this case, a(n) represents the coefficients of the terms involving powers of x in the power series representation of f(x). Here, the coefficients are always non-zero, except for the zero terms after x^3.

Since all the non-zero coefficients for powers of x in the power series representation are constant, the ratio |a(n)/a(n+1)| is a constant as well. Hence, the limit of the ratio is also constant, which means the radius of convergence is infinite, or R = ∞.

Therefore, the power series representation of f(x) converges for all real numbers, meaning it converges everywhere.