(a)
To find: The Taylor series.
(a)
Answer to Problem 60E
Explanation of Solution
Given information:
The function is f(x) = In (1 + x) at x = 0.
Formula used:
Maclaurin series is used.
Calculation:
Form the list of Maclaurin series
Conclusion:
(b)
To find: The value of x the series converges.
(b)
Answer to Problem 60E
The series converges for
Explanation of Solution
Given information:
The value of x does the series converges
Formula used:
The ratio Test is used.
Calculation:
The series of absolute values is
The ratio Test: Let
Then, the series converges if L < I, the series diverges if L >1 and the test is inconclusive if
L =1.
Using the ratio test, we check for absolute convergence as follows,
The series converges absolutely for
And diverges for
Check at
Check at
The series converges for
Conclusion:
The series converges for
(c)
The first five non-zero terms.
(c)
Answer to Problem 60E
A bound for the (absolute) truncation error is 0.002605
Explanation of Solution
Given information:
To bound for the error in evaluating In (3/2) by using only the first five nonzero terms of the series in part (a).
Formula used:
Calculation:
To estimate
The truncation error is less than the magnitude of the sixth nonzero term, that is
Thus, a bound for the (absolute) truncation error is 0.002605
Conclusion:
A bound for the (absolute) truncation error is 0.002605
(d)
The logarithmic function
(d)
Answer to Problem 60E
The logarithmic function is
Explanation of Solution
Given information:
To determine the logarithmic function whose Taylor series is
Formula used:
Taylor series is used.
Calculation:
The series is
To determine the logarithmic function as
Conclusion:
The logarithmic function is
Chapter 10 Solutions
Calculus: Graphical, Numerical, Algebraic
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