- 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{equation} \frac{1}{(1 - x)} = 1 + x + x^ 2 + \ldots + x^{n-1} + \frac{x^{n}}{(1 - x)}\label{eq:32jb} \end{equation}43and compare the ”remainder”
\begin{equation} ∫_ 0^ x\frac{t^{n}}{(1 - t)}\, {\mathbb {d}}t\label{eq:resto_ log_ strano_ int} \end{equation}44thus obtained with with the "integral remainder" of \(f(x) = -\log (1 - x)\) (as presented in Exercise [1BR]).
Proceed similarly for \(\arctan (x)\), integrating
\begin{equation} 1/(1 + x^ 2 ) = 1 - x^ 2 + x^ 4 + \ldots + (-1)^{n} x^{2n} - (-1)^ n x^{2n+2} /(1 + x^ 2 )\quad .\label{eq:2e98a} \end{equation}45Solution 1
EDB — 1FR
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"Mennucci , Andrea C. G."
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