1. Consider the series ∞

X n=3

loge n np

, p ≥ 1.

Determine the values of p for which the series converges. 2. Consider the integral Z π 4 0 cos(x2)dx. Since we cannot evaluate the integral exactly, we will approximate it using Maclaurin polynomials.

(a) Determine P4(x), the 4th degree Maclaurin polynomial of the integrand cos(x2). (b) Use MATLAB to check your answer to part (a). (c) Find an approximation to the integral by integrating P4(x). (d) Obtain an upper bound on the magnitude of the error in the integration in part (c).

3. Consider the periodic function

g(t) =

0, −1 < t < 0

cos(πt), 0 < t < 1

with g(t) = g(t + 2).

(a) Determine a general expression for the Fourier series of g. (b) Use MATLAB to plot both g and the sum of the ﬁrst 5 non-zero terms of the Fourier series for g on the same set of axes for −3 < t < 3.

2019-01-19T08:37:26+00:00
Maths