Finding the Maclaurin series
Maclaurin series as the Taylor series centered around a=0
A Maclaurin series is the specific instance of the Taylor series when ???a=0???.
Remember that we can choose any value of ???a??? in order to find a Taylor polynomial. Maclaurin series eliminate that choice and force us to choose ???a=0???.
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Remember that we would always use the formula
???\frac{f^{(n)}(a)}{n!}(x-a)^n???
to build each term in the Taylor series. Since ???a=0??? in every Maclaurin series, this formula simplifies to
???\frac{f^{(n)}(0)}{n!}(x-0)^n???
???\frac{f^{(n)}(0)}{n!}x^n???
Everything else about the Maclaurin series is the same.
How to build the Maclaurin series
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Finding the nth-degree Maclaurin series
Example
Find the seventh-degree Maclaurin series of the function.
???f(x)=\sin{(3x)}???
We’ll start by creating the chart we’ve always made for Taylor polynomials. Since we’re finding the series to the seventh-degree, we’ll use ???n??? from ???0??? to ???7???. Since it’s a Maclaurin series, we’ll use ???a=0???.
Maclaurin series eliminate the TAYLOR SERIES choice and force us to choose a=0.
With the whole chart filled in, we can build each term of the Maclaurin series.
Putting all of the terms together, we get the seventh-degree Maclaurin series.
???0+3x+0-\frac{27}{6}x^3+0+\frac{81}{40}x^5+0-\frac{243}{560}x^7???
???3x-\frac{27}{6}x^3+\frac{81}{40}x^5-\frac{243}{560}x^7???
