Search for question

(a) A damping component D of a circuit is a component that affects an input signal by diminish-

ing its strength by M%, where 0 < M < 100, such that, for an input a, the output will be

a-pa, where p = M/100. Let us consider a circuit with n such components D₁, D₂

as presented in the figure below.

D₁

a-pa

02

01

D₂

Consider the sequence (a) defined such that a,, is the value of the signal just before the

7th component.

i. Give the expression for a₁, ₂ and as in terms of a and r defined as r = 1 - p.

[1 marks]

(D₂₁)

an

ii. What type of sequence is (a,,)?

[1 marks]

iii. Prove that for a strictly positive integer N the partial sum Sy associated with this

1-

sequence is given by Sn = a

What is the limit of this sum as N tends to

infinity?

1-

[3 marks]

(b) Let us now consider the function f(r)=for a constant a 0.

d" f

i. By inspection, find an expression form for any integer n > 0.

[1 marks]

ii. Use this to write down the Taylor polynomial PN (r) for some strictly positive integer N

about the point r = 0.

[1 marks]

iii. Relate this Taylor polynomial to the partial sum studied in (a) and use this relationship

to compute the remainder term Ry(r) occurring in Taylor's theorem. By studying

the behaviour of Ry(r) as roo, show that f(r) is equal to its Taylor series for

0

[3 marks]

[Total: 10 marks]

Fig: 1