Series

Mathematical Modelling & Analysis II Coursework I:

Series and Transforms Page 1

ENGS203P Coursework I: Series and Transforms

1. Evaluate the sum of the following series: ¦ ¸

¹

· ¨

©

999 §

1 10

1 log n n

n

[Hint: write out first 3 or 4 terms and last 3 or 4 terms] [6]

2. Show that the first four terms of the Maclaurin series expansion of y(x), where loge (y) = e

x

, are as follows:

¸

¸

¹

·

¨

¨

©

§ … 6

5 1

3

2 x

y e x x [10]

Calculate the percentage error between the true value of y and that derived from the first four terms of the above

expansion when x = 0.3 . [4]

3. A voltage v(t) is applied to a coil to switch on a magnetic field periodically for an interval of S seconds. The

voltage as a function of time t is described by the function below:

v(t) = 4 for 0 < t < S/2

v(t) = í4 for -S < t <

v(t) = 0 for -S < t < -S and S < t < S

a) Sketch v(t) over the interval from t = -S to t = +S. If this function v(t) is assumed to be periodic over an

interval of 2S, it is possible to express the function as a Fourier series as follows:

+ ( a mt+ b mt) 2

a

v t = m m

m=1

0 ( ) ¦ cos sin

f

. [4]

b) Use the Fourier series integrals to show that:

am 0 for all values of m (including m = 0), and that:

> @ 1 cos( / 2) 8

S

S

m

m

bm . [20]

c) Thus show that v(t) can be represented by the following series:

¸

¹

· ¨

©

§ sin7 … 7

1 sin6

3

1 sin5

5

1 sin3

3

1 sin sin 2 8

v(t) t t t t t t

S

. [6]

d) Use Matlab to plot the above series expansion of v(t) over the interval from t = -S to t = +S using only the first

two, three, four, five, six, and one hundred terms (i.e. plot six graphs on the same axes). Briefly describe (in

words) the effect of including an increasing number of terms in the series. [10]

e) It can be shown that the Fourier transform of v(t) is given by: >1 cos( )@ 4 ( ) 2

u

u

i V u S

S .

By representing V(u) as a complex number of the form p í iq, evaluate p and q at values of u = n/2S for

integer values of n in the range 0 o 7. Hence sketch q over the interval from u = 0 to u = 7/2S. [14]

f) By comparing the values of p and q found in part (e) with the values of am and bm found in part (c), explain

the relationship between the Fourier transform of a periodic function and the coefficients of its Fourier series.

[6]

4. Solve the following linear second-order differential equation a) using Laplace transforms, and b) by calculating

the complementary function and (be careful!) the particular integral:

x y e

dx

dy

dx

d y 2 2

2

[20]

Calculate both solutions (which should be the same) using the initial condition that y = 0 and dy/dx = 1 at x = 0.

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