Calculate the following: a. Find the Maclaurin series of cos⁡(x) and find the radius of this series, without using any known power or Maclaurin series, besides geometric. b. Find exactly the series of cos⁡(−2x)

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diskusje5 · Accepted Answer

(a) The Maclaurin series of a function f(x) is given byf(x)=f(0)+f′(0)x+f″(0)x22!+f‴(0)x33!+f(4)(0)x44!+...We have f(x)=cos⁡xCompute the first few derivatives of cos⁡x.Now, f(0)=1f′(0)=0f″(0)=−1f‴(0)=0f(4)(0)=1So, the Maclaurin series iscos⁡x=1−x22!+x44!±...=1+(−1)x2⋅12!+(−1)2x2⋅24!+...⇒cos⁡x=∑n=0∞(−1)nx2n(2n)! is the required Maclaurin series.Use Ratio Test for radius of convergence:limn→∞|(−1)n+1x2(n+1)(2(n+1))!(−1)nx2n(2n)!|=limn→∞|x2n+2x2n⋅(2n)!(2n+2)!|=x2limn→∞|(2n)!(2n+2)⋅(2n+1)⋅(2n)!|=x2limn→∞|1(2n+2)⋅(2n+1)|=0&amp;lt;1The series converges independently of the value of x. Hence, the radius of the series is ∞ (b) We have f(x)=cos⁡(−2x)Compute the first few derivatives of cos x.f′(x)=2sin⁡(−2x)f″(x)=−22cos⁡(−2x)f‴(x)=−23sin⁡(−2x)f(4)(x)=24cos⁡(−2x)Now,f(0)=1f′(0)=0f″(0)=−22f‴(0)=0f(4)(x)=24So, the Maclaurin series iscos⁡(−2x)=1−22x22!+24x44!±...=1+(−1)22⋅1x2⋅12!+(−1)222⋅2x2⋅24!+...

Jeffrey Jordon · Accepted Answer

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Calculate the following:a. Find the Maclaurin series of cos(x) and find the radius of this series, without using any known power or Maclaurin series, besides geometric.b. Find exactly the series of cos(-2x)

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