Find the Maclaurin series for f(x)=x^4cos x. Also, find the third term for this series. The Maclaurin series for cos(x) is given by: cos(x)=sum_{n=0}^infty(-1)^nfrac{x^{2n}}{(2n)!}=1-frac{x^2}{2!}+frac{x^4}{4!}-frac{x^6}{6!}+frac{x^8}{8!}-frac{x^{10}}{10!}+...

aortiH 2021-02-02 Answered
Find the Maclaurin series for f(x)=x4cosx. Also, find the third term for this series.
The Maclaurin series for cos(x) is given by:
cos(x)=n=0(1)nx2n(2n)!=1x22!+x44!x66!+x88!x1010!+...
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Expert Answer

lamanocornudaW
Answered 2021-02-03 Author has 85 answers

The Maclaurin series for cos(x) is given by:
cos(x)=n=0(1)nx2n(2n)!=1x22!+x44!x66!+x88!x1010!+...
Multiply the Maclaurin series for cos(x) by x4 and obtain the Maclaurin series for f(x)=x4cos(x) as shown below.
x4cos(x)=x4(n=0(1)nx2n(2n)!)
=n=0(1)nx4x2n(2n)!
=n=0(1)nx2n+4(2n)!
Thus, the Maclaurin series for f(x)=x4cos(x) is,
f(x)=n=0(1)nx2n+4(2n)!
Now find the third term for this series as shown below.
f(x)=n=0(1)nx2n+4(2n)!
=x4x62!+x84!x106!+x128!...
=x4x62+x824x10720+x1240320...
Hence, the third term of the series is x824

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Jeffrey Jordon
Answered 2021-12-17 Author has 2262 answers

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