The Maclaurin series is a special case of the more general Taylor series. The Taylor series of f expanded about x = a is given by
and hence is expanded in powers of x - a.
For Taylor series, enter the number of terms and the point of expansion in the Series dialog box. To find the Taylor series of ln x expanded about x = 1, choose Powers Series. In the dialog box, select the desired number of terms and expand about the point x - 1.
Power Series
ln x =x - 1
-
x - 1
+
x - 1
-
x - 1
+ O
x - 1
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A comparison between ln x and the polynomial
x - 1
-
x - 1
+
x - 1
-
x - 1
is illustrated graphically in the
following figure.
Plot 2D + Rectangular
ln x,x - 1
-
x - 1
+
x - 1
-
x - 1
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dtbpF3in2.0003in0pt
You can produce the following power series expansions with Scientific Notebook.
Power Series
=
-
x - 2
+
x - 2
-
x - 2
+
x - 2
+ O
x - 2
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sin x = -x - π
+
x - π
+ O
x - π
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= 1 +
x - 1
-
x - 1
+
x - 1
-
x - 1
+ O
x - 1
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csc x = 1 +x -
π
+
x -
π
+ O
x -
π
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2 sin2x = 2x - π
-
x - π
+ O
x - π
![]()
1 - cos 2x = 2x - π
-
x - π
+ O
x - π
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