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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