Representing a function as a Power Series. Suppose that we want to represent the function f(x) = In(5z + 10) as a power series centered at zero. We can start by finding the power series representation (centered at zero) for the derivative of the function, f'(x). The power series representation (centered at zero) for f'(x) is: fla) - Σ (-1)"(음) n+1 x" 5 The series you found above converges on the interval Next, integrate term by term the series you found. This will enable you to write the series representation for f(x) = In(5x + 10). %3| In (sr + 10) □+ (-1)씨 )( n+1 n+1 Note: the second to last blank above is for the constant of integration can which be found by substituting z = 0 into both sides of the equation containing the antidifferentiated series.
Representing a function as a Power Series. Suppose that we want to represent the function f(x) = In(5z + 10) as a power series centered at zero. We can start by finding the power series representation (centered at zero) for the derivative of the function, f'(x). The power series representation (centered at zero) for f'(x) is: fla) - Σ (-1)"(음) n+1 x" 5 The series you found above converges on the interval Next, integrate term by term the series you found. This will enable you to write the series representation for f(x) = In(5x + 10). %3| In (sr + 10) □+ (-1)씨 )( n+1 n+1 Note: the second to last blank above is for the constant of integration can which be found by substituting z = 0 into both sides of the equation containing the antidifferentiated series.
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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Transcribed Image Text:Representing a function as a Power Series.
Suppose that we want to represent the function f(x) = In(5z + 10) as a power series centered at zero. We can start by finding the power series
representation (centered at zero) for the derivative of the function, f'(x).
The power series representation (centered at zero) for f'(x) is:
fla) - Σ (-1)"(음)
n+1
x"
5
The series you found above converges on the interval
Next, integrate term by term the series you found. This will enable you to write the series representation for f(x) = In(5x + 10).
%3|
In (sr + 10) □+ (-1)씨 )(
n+1
n+1
Note: the second to last blank above is for the constant of integration can which be found by substituting z = 0 into both sides of the equation
containing the antidifferentiated series.
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