Question

1. Find the center, radius and interval of convergence of the following series

a) sum n = 1 to ∞ (3 ^ (n + 1) * (x - 7) ^ n)/n

b) sum n = 1 to ∞ (2 * (4x + 1) ^ n)/(n ^ 2)

d) sum n = 1 to ∞ ((- 1) ^ (n + 1) * (x - 1) ^ n)/n

e) sum n = 1 to ∞ (2 ^ n * (4x - B) ^ n)/n

c) sum n = 1 to ∞ ((- 1) ^ n * (x - 2) ^ (2n + 1))/n

2. Find the power series representation for the following functions and determine their interval of convergence

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

b) f(x) = (x ^ 2)/(a ^ 3 + x ^ 3)

3.) Find the Maclaurine series of the following functions.

a) f(x) = e ^ (3x)

b) f(x) = 2x ^ 3 * e ^ (- 2x ^ 2)

4. Use the binomial series and expand the given function as power series.

a. f(x) = 1/((1 + x) ^ 3)

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

c. f(x) = (2x - 3) ^ 4

Answer 1

For each part (a-e), you need to determine the center, radius, and interval of convergence of the given power series. This involves using the ratio test or root test to check for convergence and applying the formulas for the center and radius of convergence.

In part (a), you need to find the power series representation of the function (x^3)/(x+2) by expressing it as a geometric series and finding its interval of convergence. Similarly, in part (b), you need to represent the function (x^2)/(a^3 + x^3) as a power series.

For each part (a-b), you need to find the Maclaurin series expansion of the given functions. This involves finding the derivatives of the function at x = 0 and using the formulas for the Maclaurin series coefficients.

In each part (a-c), you are required to expand the given function using the binomial series. This involves applying the binomial theorem to expand the function as a power series.

It is recommended to use mathematical software or a programming language such as MATLAB or Python to perform the necessary calculations for these problems.