Determining first four non-zero terms of fourier series

Question

1. A. Consider the limit

lim_n→∞  ((n + p)! + pⁿ + np + log_e np)/n! where p is a fixed integer, p∈Z. Determine the values of p, such that the limit exists. Carefully indicate any Standard Limits that you use.

B. Consider the series ^∞∑_n=0 cos (mnπ))/(n +1) where m is a fixed integer, m ∈ Z. Determine the values of m, such that the series converges. Explain your reasoning in detail.

2. Consider the function f(x) = sin[π/2 + x]

A. Determine the sixth-order Taylor polynomial  or f(x) about x = π/2 .

B. Write an expression for the error |R6(x)|.

C. Estimate |R6(x)| on the interval |x| ≤  0.2.

D. Verify your answer in part 2A using MATLAB.

3. Consider the function f(x) = |sin x|; -π ≤x ≤ π; f(x) = f(x + 2π)

A. Determine the first 4 non-zero terms of the Fourier series for f(x).

B. Conjecture the form of the n-th term of the Fourier series for f(x).

C. Use MATLAB to plot the first two non-zero terms of f(x), then the first four non-zero terms of f(x), and finally the full function f(x), on the same set of axes (so that all 3 results are in the same figure).

Full working must be shown in your analytic solutions.