- #1
jsewell94
- 23
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Homework Statement
The first three terms of a Taylor Series centered about 1 for [itex]ln(x)[/itex] is given by:
[itex]\frac{x^{3}}{3}[/itex] - [itex]\frac{3x^{2}}{2}[/itex] + [itex]3x[/itex] - [itex]\frac{11}{6}[/itex]
and that
[itex]\int{ln(x)dx}[/itex] = [itex]xlnx - x + c[/itex]
Show that an approximation of [itex]ln(x)[/itex] is given by:
[itex]\frac{x^3}{12}[/itex] - [itex]\frac{x^2}{2}[/itex] + [itex]\frac{3x}{2}[/itex] - [itex]\frac{5}{6}[/itex] - [itex]\frac{1}{4x}[/itex]
2. The attempt at a solution
I have tried this problem a few times, but it is becoming clear that I am missing some crucial step/idea. Basically, what I have tried is setting lnx equal to the Taylor Series, integrating both sides, and solving for lnx. However, when I do this, I manage to get all of the necessary terms EXCEPT for the 1/4x. Where does that come from, exactly? If someone could help, that'd be awesome! :D
Thanks!