in effect, any partial sum of the series is a suitable Taylor polynomial of the function f(x) at zero. The formula (2.1) is often used to approximate the values of the natural logarithm. Say, setting x = 1/3, we obtain from (2.1) the infinite series representing In(2), due to 1+1/3 1-1/3 = 2. (i) Find a rational number x = p/q which is the solution of the equation
in effect, any partial sum of the series is a suitable Taylor polynomial of the function f(x) at zero. The formula (2.1) is often used to approximate the values of the natural logarithm. Say, setting x = 1/3, we obtain from (2.1) the infinite series representing In(2), due to 1+1/3 1-1/3 = 2. (i) Find a rational number x = p/q which is the solution of the equation
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:(Taylor Series/Polynomials, Natural Logarithm). All numerical answers, unless otherwise required, should be rounded to 6-
digit floating-point numbers.
(0) It is known that
for all x € (-1, 1).
x =
and enter it (in reduced form) in the input field below:
In fact, the series in the right-hand side is the Taylor series at zero of the function
x²k+1\
x
In 1+3 = 2 (2 32++1)
1-x
k=U
in effect, any partial sum of the series is a suitable Taylor polynomial of the function f(x) at zero.
The formula (2.1) is often used to approximate the values of the natural logarithm. Say, setting x = 1/3, we obtain from
(2.1) the infinite series representing In(2), due to
1 + 1/3
1-1/3
(i) Find a rational number x = p/q which is the solution of the equation
1 + x
1-x
Find an approximation y* of y = In(1.38)
*
f(x) = In
RE(y y*) =
1+x
SD(y = y*) =
- X
We then have In(1.38) is equal to the sum of the series in (2.1) for x = p/q.
(ii) Consider the first five terms of the series in (2.1), that is, the ninth Taylor polynomial of the function f(x) at zero,
as
= 2.
= 1.38
TX) = 2(x + = + = + = + =)
T(x)
* = T(p/q),
where p/q is the rational number you have found in (i), and enter your result, rounded to a 6-digit floating-number, in the
input field below:
(iii) Find the absolute, the relative error, and the number of significant digits in the approximation of y = ln(1.38)
AE(y y*) =
(2.1)
by y* :
Transcribed Image Text:and enter it (in reduced form) in the input field below:
X =
We then have In(1.38) is
(ii) Consider the first fiv
Find an approximation y* of y = In(1.38) as
=
T(x) = 2 ( x + ² + ² + ² + )
3
5
7
9
y* = T(p/q),
where p/q is the rational number you have found in (i), and enter your result, rounded to a 6-digit floating-number, in the
input field below:
RE(y≈ y*) =
ion f(x) at zero,
(iii) Find the absolute, the relative error, and the number of significant digits in the approximation of y = In(1.38) by y*:
AE(y = y*) =
SD(y = y) =
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