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Hatshy [7]
3 years ago
12

Find the first four nonzero terms of the taylor series about 0 for the function f(x)=x4sin(6x). note that you may want to find t

hese in a manner other than by direct differentiation of the function.
Mathematics
1 answer:
ziro4ka [17]3 years ago
4 0

The quickest way to get the series is if you already know the series for sine:

\sin6x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n(6x)^{2n+1}}{(2n+1)!}

Then just multiply the series by x^4 to get

f(x)=x^4\sin6x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n6^{2n+1}x^{2n+5}}{(2n+1)!}

The first four nonzero terms are simply the first four terms:

6x^5-\dfrac{6^3}{3!}x^7+\dfrac{6^5}{5!}x^9-\dfrac{6^7}{7!}x^{11}

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