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Find the Taylor polynomial for 1 / (1+x)

by RoRi

Show that

    \[ T_n \left( \frac{1}{1+x} \right) = \sum_{k=0}^n (-1)^k x^k. \]

Starting with the algebraic identity

    \begin{align*} \frac{1}{x+1} &= 1 - x + x^2 - x^3 + \cdots + (-1)^n x^n + (-1)^{n+1} \frac{x^{n+1}}{1+x} \\[9pt] &= P_n (x) + x^n \frac{(-1)^n x}{1+x} \end{align*}

Since \frac{(-1)^n x}{1+x} \to 0 as x \to 0 we may apply Theorem 7.4 (page 277 of Apostol) to conclude

    \[ T_n \left( \frac{1}{1+x} \right) = P_n (x) = \sum_{k=0}^n (-1)^k x^k. \]

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