Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY ALGEBRA

1.16 Express the following in partial fraction form:

(a)

2 x^3 − 5 x+1

x^2 − 2 x− 8

, (b)

x^2 +x− 1

x^2 +x− 2

.

1.17 Rearrange the following functions in partial fraction form:

(a)

x− 6

x^3 −x^2 +4x− 4

, (b)

x^3 +3x^2 +x+19

x^4 +10x^2 +9

.

1.18 Resolve the following into partial fractions in such a way thatxdoes not appear
in any numerator:

(a)

2 x^2 +x+1

(x−1)^2 (x+3)

, (b)

x^2 − 2

x^3 +8x^2 +16x

, (c)

x^3 −x− 1

(x+3)^3 (x+1)

.

Binomial expansion

1.19 Evaluate those of the following that are defined: (a)^5 C 3 ,(b)^3 C 5 ,(c)−^5 C 3 ,(d)

− (^3) C 5.
1.20 Use a binomial expansion to evaluate 1/

√

4 .2 to five places of decimals, and

compare it with the accurate answer obtained using a calculator.

Proof by induction and contradiction

1.21 Prove by induction that

∑n

r=1

r=^12 n(n+1) and

∑n

r=1

r^3 =^14 n^2 (n+1)^2.

1.22 Prove by induction that

1+r+r^2 +···+rk+···+rn=

1 −rn+1

1 −r

.

1.23 Prove that 3^2 n+7, wherenis a non-negative integer, is divisible by 8.
1.24 If a sequence of terms,un, satisfies the recurrence relationun+1=(1−x)un+nx,
withu 1 = 0, show, by induction, that, forn≥1,

un=

1

x

[nx−1+(1−x)n].

1.25 Prove by induction that

∑n

r=1

1

2 r

tan

(

θ

2 r

)

=

1

2 n

cot

(

θ

2 n

)

−cotθ.

1.26 The quantitiesaiin this exercise are all positive real numbers.

(a) Show that

a 1 a 2 ≤

(

a 1 +a 2

2

) 2

.

(b) Hence prove, by induction onm,that

a 1 a 2 ···ap≤

(

a 1 +a 2 +···+ap

p

)p

,

wherep=2mwithma positive integer. Note that each increase ofmby

unity doubles the number of factors in the product.