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E42

[1FR] Prerequisites:[1BR].Note:From an idea in Apostol’s book [ 5 ] , Chapter 7.3.Write Taylor’s polynomial (around $x_{0} = 0$ ) for $- \log (1 - x)$ , integrating

\begin{matrix} (1) & \frac{1}{(1 - x)} = 1 + x + x^{2} + \dots + x^{n - 1} + \frac{x^{n}}{(1 - x)} \end{matrix}

and compare the ”remainder”

\begin{matrix} (2) & \int_{0}^{x} \frac{t^{n}}{(1 - t)} d t \end{matrix}

thus obtained with with the "integral remainder" of $f (x) = - \log (1 - x)$ (as presented in Exercise [1BR]).

Proceed similarly for $\arctan (x)$ , integrating

\begin{matrix} (3) & 1 / (1 + x^{2}) = 1 - x^{2} + x^{4} + \dots + (- 1)^{n} x^{2 n} - (- 1)^{n} x^{2 n + 2} / (1 + x^{2}) . \end{matrix}

Solution 1

[1FS]

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Authors: "Mennucci , Andrea C. G." .

Bibliography
Book index

Taylor's theorem, with integral remainder
Taylor's theorem

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