Asymptotic sequence and expansions

Lesson 6

Asymptotic sequence and expansions – p. 1/15

Definition of asymptotic expansionas x → ∞

The expansion f(x) = a0 + a1

x + a2

x2 + . . . + aN

xN + RN is anasymptotic expansion if for any N

RN = O(1

xN+1) x → ∞

In this case we write

f(x) ∼∞

∑

n=0

an

xnas x → ∞ (not n → ∞)

limN→∞

RN = ∞ ∀x fixed

limx→∞

RN = 0 ∀N fixedAsymptotic sequence and expansions – p. 2/15

Finding the coefficients

• How to find a0, a1, a2, . . .?

Since f(x) = a0 + O(

1x

)

, then a0 = limx→∞

f(x)

• From f(x) = a0 + a1

x + O(

1x2

)

we get

x(f(x) − a0) = a1 + O(

1x

)

. So

a1 = limx→∞

x(f(x) − a0)

Asymptotic sequence and expansions – p. 3/15

Finding the coefficients

• How to find a0, a1, a2, . . .?

Since f(x) = a0 + O(

1x

)

, then a0 = limx→∞

f(x)

• From f(x) = a0 + a1

x + O(

1x2

)

we get

x(f(x) − a0) = a1 + O(

1x

)

. So

a1 = limx→∞

x(f(x) − a0)

Asymptotic sequence and expansions – p. 3/15

Finding the coefficients

• Further, from f(x) = a0 + a1

x + a2

x2 + O(

1x3

)

a2 = limx→∞

x2(

f(x) − a0 −a1

x

)

• In general

aN = limx→∞

xN(

f(x) −N−1∑

n=0

an

xn

)

• {1, 1x , 1

x2 ,1x3 , . . .} is an asymptotic sequence

Asymptotic sequence and expansions – p. 4/15

Finding the coefficients

• Further, from f(x) = a0 + a1

x + a2

x2 + O(

1x3

)

a2 = limx→∞

x2(

f(x) − a0 −a1

x

)

• In general

aN = limx→∞

xN(

f(x) −N−1∑

n=0

an

xn

)

• {1, 1x , 1

x2 ,1x3 , . . .} is an asymptotic sequence

Asymptotic sequence and expansions – p. 4/15

Finding the coefficients

• Further, from f(x) = a0 + a1

x + a2

x2 + O(

1x3

)

a2 = limx→∞

x2(

f(x) − a0 −a1

x

)

• In general

aN = limx→∞

xN(

f(x) −N−1∑

n=0

an

xn

)

• {1, 1x , 1

x2 ,1x3 , . . .} is an asymptotic sequence

Asymptotic sequence and expansions – p. 4/15

Asymptotic sequences

If we write ε = 1x for x ∼ ∞, then

{1, ε, ε1, ε2, ε3, . . .}

{1, ε1/2, ε1, ε3/2, ε2, . . .}

en general if δn(ε) is a term of asymptotic sequencethen

{δ0(ε), δ1(ε), δ2(ε), . . .}

δn+1(ε) = o(δn(ε)) ε → 0

Asymptotic sequence and expansions – p. 5/15

Examples of asymptotic sequences

{1, sin ε, sin2 ε, sin3 ε, . . . , sinn ε, . . .}

{1, ln(1 + ε), ln(1 + ε2), . . . ln(1 + εn), . . .}

{tan(ε1/2), tan(ε), tan(ε3/2), . . . , tan(εn/2), . . .}

{ln ε

ε,1

ε, ln ε, 1, ε ln ε, ε, ε2 ln ε, ε2, . . .}

Asymptotic sequence and expansions – p. 6/15

Expansion in terms of δn(ε)

f(ε) ∼∑

∞

n=0 anδn(ε) that means

f(ε) =N

∑

n=0

anδn(ε) + RN , RN = O(δn+1(ε)) ε → 0

aN = limε→0

(f(ε) −∑N−1

n=0 anδn(ε)

δN (ε)

)

Asymptotic sequence and expansions – p. 7/15

Expansion in terms of δn(ε)

f(ε) ∼∑

∞

n=0 anδn(ε) that means

f(ε) =N

∑

n=0

anδn(ε) + RN , RN = O(δn+1(ε)) ε → 0

aN = limε→0

(f(ε) −∑N−1

n=0 anδn(ε)

δN (ε)

)

Asymptotic sequence and expansions – p. 7/15

Positive example

Let us decompose the function eε usingδn(ε) = ln(1 + εn), e. g.

eε ∼ a0 + a1 ln(1 + ε) + a2 ln(1 + ε2) + . . .

a0 = 1

eε − 1 = a1 ln(1 + ε) + o(ln(1 + ε)),

limε→0

( eε − 1

ln(1 + ε)

)

= a1 + limε→0

(o(ln(1 + ε))

ln(1 + ε)

)

a1 = limε→0

( eε − 1

ln(1 + ε)

)

= limε→0

(1 + ε + O(ε2) − 1

ε + O(ε2)

)

= 1

Asymptotic sequence and expansions – p. 8/15

Positive example

eε − 1 − ln(1 + ε) = a2 ln(1 + ε2) + o(ln(1 + ε2)),

a2 = limε→0

(eε − 1 − ln(1 + ε)

ln(1 + ε2)

)

= limε→0

(1 + ε + ε2/2 + O(ε3) − 1 − ε + ε2/2

ε2 + O(ε3)

)

= 1

eε = 1 + ln(1 + ε) + ln(1 + ε2) −1

6ln(1 + ε3) + o(ln(1 + ε3))

Asymptotic sequence and expansions – p. 9/15

Negative example

Let us attempt to expand cos(ε1/2 + ε) using the sameasymptotic equation δn(ε) = ln(1 + εn)

cos(ε1/2 + ε) ∼ a0 + a1 ln(1 + ε) + a2 ln(1 + ε2) + . . .

a0 = cos(ε1/2 + ε) = 1

a1 = limε→0

(cos(ε1/2 + ε) − 1

ln(1 + ε)

)

= limε→0

(1 − (ε1/2 + ε)2/2 + O(ε2) − 1

ε + O(ε2)

)

= −1

2

Asymptotic sequence and expansions – p. 10/15

Negative example

a2 = limε→0

(cos(ε1/2 + ε) − 1

ln(1 + ε)

)

= limε→0

(1 − (ε + 2ε3/2 + ε2)/2 + O(ε2) − 1 + ε/2 − ε2/4

ε2 + O(ε4)

)

= limε→0

(

−ε3/2 + O(ε2)

ε2 + O(ε4)

)

= −∞

What is the solution of the problem? Add the termln(1 + ε3/2) and its subsequent powers.

Asymptotic sequence and expansions – p. 11/15

Uniqueness of expansions

• Any function f(x) produce unique expansion.

• Let us consider f1 = sin ε andf2 = sin ε + exp(−1/ε2). If

sin ε + exp(−1/ε2) = a0 + a1ε + a2ε2 + a3ε

3 + . . .

then a0 = 0,

a1 = limε→0

(sin ε + exp(−1/ε2)

ε

)

= limε→0

(sin ε

ε

)

+ limε→0

(+ exp(−1/ε2)

ε

)

= 1 + 0 = 1

Asymptotic sequence and expansions – p. 12/15

Uniqueness of expansions

• Any function f(x) produce unique expansion.• Let us consider f1 = sin ε and

f2 = sin ε + exp(−1/ε2). If

sin ε + exp(−1/ε2) = a0 + a1ε + a2ε2 + a3ε

3 + . . .

then a0 = 0,

a1 = limε→0

(sin ε + exp(−1/ε2)

ε

)

= limε→0

(sin ε

ε

)

+ limε→0

(+ exp(−1/ε2)

ε

)

= 1 + 0 = 1

Asymptotic sequence and expansions – p. 12/15

Uniqueness of expansions

a2 = limε→0

(sin ε + exp(−1/ε2) − ε

ε2

)

= 0

a3 = limε→0

(sin ε + exp(−1/ε2) − ε

ε3

)

= −1

6

exp(−1/ε2) = 0 · 1 + 0 · ε + 0 · ε2 + 0 · ε3 + . . .

Asymptotic sequence and expansions – p. 13/15

Taylor series is asymptoticexpansion

• Why the Taylor series at 0 are asymptoticexpansions?

• f(x) =∑N

n=0 anxn + RN with

RN =xN+1

(N + 1)!f (N+1)(z).

SoRN = O(xN+1) x → 0.

limx→0

RN

xN+1=

1

(N + 1)!f (N+1)(0) = 0

Asymptotic sequence and expansions – p. 14/15

Taylor series is asymptoticexpansion

• Why the Taylor series at 0 are asymptoticexpansions?

• f(x) =∑N

n=0 anxn + RN with

RN =xN+1

(N + 1)!f (N+1)(z).

SoRN = O(xN+1) x → 0.

limx→0

RN

xN+1=

1

(N + 1)!f (N+1)(0) = 0

Asymptotic sequence and expansions – p. 14/15

The end

Asymptotic sequence and expansions – p. 15/15