1. A. Consider the limit
lim_{n→∞} ((n + p)! + p^{n} + 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. describe 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. prepare 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.