Understanding Engineering Mathematics

3.3.10 Infinite series

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A.Identify which are geometric sequences, and sum them to infinity.

(i) 1, 2 , 3 , 4 ,... (ii) − 1 , 3 , 5 , 9 ,...

(iii) 1, 1 , 1 , 1 ,... (iv)^12 ,^13 ,^14 ,^15 ,...

(v) 1,

1

3

,

1

9

,

1

27

,... (vi) 0. 1 , 0. 3 , 0. 5 , 0. 7 ,...

(vii) 2,− 4 , 8 ,− 16 ,... (viii) 0. 2 , 0. 04 , 0. 008 , 0. 0016 ,...

B.Find the sum of the infinite geometric series with first terms and common ratios given

respectively by

(i) 1, 2 (ii) 2,^12

(iii) − 1 ,1(iv)1,^13

3.3.11 Infinite binomial series

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Find the first four terms of the binomial expansion of the following:

(i) ( 1 +x)−^1 (ii) ( 1 − 3 x)−^1

(iii) ( 1 + 4 x)−^2 (iv) ( 1 −x)

1

2

3.4 Applications

1.For each of the following functions, obtain simplest forms of the expressions for

(a) f(a+h)−f(a) (b)

f(a+h)−f(a)

h

(i) 2x (ii) x^2 (iii) 4x^3 − 6 x^2 + 3 x− 1

In each case give the result when you lethtend to zero. These sorts of calculations

are required indifferentiation from first principles(Chapter 8).

2.In statistics themean,x ̄,andvariance,σ^2 ,ofasetofnnumbersx 1 ,x 2 ,...,xnare

defined by

x ̄=

1

n

∑n

i= 1

xi σ^2 =

1

n

∑n

i= 1

(xi− ̄x)^2

respectively. Show thatσ^2 may be alternatively written as

σ^2 =

1

n

(n

∑

i= 1

xi^2 −nx ̄^2

)