Mast20029 engineering mathematics assignment use matlab to

Question

Engineering Mathematics Assignment

1. Evaluate the limit

lim_n→∞ ((4/n²)-1)ⁿ

or explain carefully why it diverges. Be sure to indicate any standard limits or limit theorems as you use them.

2. Determine if the following series converges or diverges. Justify your answer with appropriate test(s).

_n=1Σ^∞(2 tanh(n) + n³/n⁵ + n log_en)

3. Let f(x) = e^-x2.

(a) Find the degree 2 Maclaurin polynomial P₂(x) for f(x).

(b) Use P₂(x) to give a rational approximation to e^-¼.

(c) Use Taylor's Theorem to give a rational upper bound for the magnitude of the error of the approximation from (b).

4. Consider the periodic function f defined according to the rule

(a) Sketch f over the range -3 < t ≤ 3.

(b) Calculate by hand the Fourier coefficients a₀ and a_n for the Fourier series F(t) for f.

(c) Use MATLAB to help compute the Fourier coefficients b_n for the Fourier series F(t). Give your answer as a simplified expression for b_n in terms of n.

You may handwrite your final expression for bn but you should include a printout showing the MATLAB parts of the calculation.

(d) Hence write down an expression for the Fourier series F(t).

(e) For which value(s) of t does the Fourier series F(t) converge to f(t)? If it does not converge to f(t), then what, if anything, does it converge to?

(f) Use MATLAB to produce a plot of the function f(t), and of the approximation to f given by the terms of the Fourier series F(t) up to and including n = 6. Plot the two on the same set of axes and over a range showing one period.