Question

Write power series representations for the functions. tan1x\tan ^{-1} x and its derivative. [Hint: Do the derivative first.]

Solution

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Answered 1 year ago
Answered 1 year ago
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First, let's find the derivative of tan1x\tan^{-1} x:

ddxtan1x=11+x2.\dfrac{\,d}{\,dx} \tan^{-1} x =\dfrac{1}{1+x^2}.

Notice that if we write this expression as

11(x2)\dfrac{1}{1-(-x^2)}

we have something similar to the geometric series that are given with

11x=n=0xn.(1)\dfrac{1}{1-x}=\sum_{n=0}^{\infty} x^n. \tag{1}

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