Taylor Series for a Composite Function

Question

The Taylor series for $$\ln(1+x)$$ centered at $$x=0$$ is $$\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$$. Which of the following is the Taylor series for $$f(x) = \ln(1+x^2)$$ centered at $$x=0$$?

Accepted Answer

$$\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{2n}}{n}$$

Answer

$$\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^n}{2n}$$

Answer

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

Answer

$$\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}2^n x^n}{n}$$

Taylor Series for a Composite Function

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