taylor approximations

**acosta0809** · August 6th 2009, 10:29 PM

the question is

Find the Taylor series expansion about x = 0 for cos^x using the series for
2 sin x cos x.

i found the series for 2 sinx cosx by using the sinx since 2 sinx cosx = sin2x

which is $2x-\frac{4x^3}{3}+\frac{4x^5}{15}+...$

but now im stuck i dont know how 2 sinx cosx can be related to cos^2x

**CaptainBlack** · August 6th 2009, 10:46 PM

Originally Posted by acosta0809

the question is

Find the Taylor series expansion about x = 0 for cos^2 x using the series for
2 sin x cos x.

i found the series for 2 sinx cosx by using the sinx since 2 sinx cosx = sin2x

which is $2x-\frac{4x^3}{3}+\frac{4x^5}{15}+...$

but now im stuck i dont know how 2 sinx cosx can be related to cos^2x

$\cos^2(x)=1-\int_0^x 2 \cos(\zeta) \sin(\zeta)\;d\zeta$

so term by term integration of the series for $2 \sin(x)\cos(x)$ can be used to find the required series

CB

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