ENGINEERING MATHEMATICS 4 POWER SERIES BA501

CHAP 2 - Power Series (Siri Kuasa)

A power series is an infinite series

where f(x) is a function in x, an represents the coefficient of the nth term, c is a constant, and x varies around c

Exponential function

The exponential function is the function . It can be defined by the

following power series

If e = number value, a. The series can be written in series:

Example: Find power series for function until 6th terms.

=

=

Example :

1. Expand the exponential function below until 4th terms:

i)

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=

=

ENGINEERING MATHEMATICS 4 POWER SERIES BA501

ii)

2. Expand the expression below as far as the 4th terms.

i)

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

ii)

iii.

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

Logarithms Function

The logarithm of a number is the exponent by which a fixed number.

Function of logarithms series is

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Exercise : i) Find the exponential series for the expression below as far as the 4th

terms.a) b)

ii) Find the coefficient of x3 in the expression below. a)

b)

(i) =

ENGINEERING MATHEMATICS 4 POWER SERIES BA501

If value of x = negative value ( exm :-x, -2x, ) , logarithm series can be written as

From logarithms equation (i) and (ii)

If value = , hence value of

equation (iv) true for all positive value of m and n

equation (iv) use for value independent x

Example :

1. Expand the function below up to 4th terms

i.

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(ii)

(iii)

(iv)

= ----

= =

=

ENGINEERING MATHEMATICS 4 POWER SERIES BA501

ii.

iii.

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

iv.

v.

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

2. Evaluate the value of ln 3 correct to 4 significant figures.

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

3. Evaluate the value of correct to 4 significant figures.

Taylor Series

A Taylor series is a series expansion of a function about a point. The expansion of a

real function about a point is given by

F unction = any function satisfying certain conditions can be expressed as a

Taylor series

Example

1 : Determine the first four terms of Taylor series for at

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=

Exercise1. Expand the function below up to 4th terms

i. ii.

ENGINEERING MATHEMATICS 4 POWER SERIES BA501

Solution :

Differentiation function

=

=

=

=

Substituting these values into Taylor Series formula:

2 : Determine the first five terms of Taylor series for at .

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

3 : Determine the Taylor Series for at as far as the terms in x4 .

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

4 : Determine the Taylor Series for at as far as first 5th terms.

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

Maclaurin Series A Maclaurin series is a Taylor series expansion of a function about

Example1 : Determine the first four terms of the Maclaurin series for .

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Exercise

1. Determine the first three terms of Taylor series for at .

( ans : )

ENGINEERING MATHEMATICS 4 POWER SERIES BA501

2 : Expand of Maclaurin Series to 4 terms.

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

3. Determine the first five terms of Maclaurin Series for .

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

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Exercise

1. Develop a series for using Maclaurin Series as far as first 5th

terms.

( ans : )

Exercise :

1. Determine the power series until first 4th terms:

i.

ii.

iii.

iv.

2. Find the expansion of Taylor series for function below at given until first 4th terms.

i. at

ii. at

3. Find the expansion of Maclaurin series for function below until first 4th terms.

i.

ii.

4. Find the value of until 4 decimal places using the expansion of

maclaurin series of

ENGINEERING MATHEMATICS 4 POWER SERIES BA501

APPENDIX

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ENGINEERING MATHEMATICS 4 POWER SERIES BA501

FORMULA

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ASAS INDEKS DAN LOGARITMA

HUKUM INDEKS1.

2.

3.

HUKUM LOGARITMA1.

2.

3.

ASAS PEMBEZAAN

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

PERSAMAAN KUADRATIK