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July 2022 1 44 Report

Разложить функцию [tex]f(x)=(1+e^{x^2})^{\frac{1}{2}}[/tex] по формуле Тейлора до [tex]o(x_4)[/tex] при [tex]x \to 0[/tex].

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igorShap

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Рассмотрим $g(t)=\sqrt{1+e^t}$ и разложим ее по формуле Тейлора до $o(t^2)$ .

$g(0)=\sqrt{2}$

$g'(t)=\dfrac{1}{2\sqrt{1+e^t}}\cdot e^t=>g'(0)=\dfrac{1}{2\sqrt{2}}\\ g''(t)=\dfrac{e^t\sqrt{1+e^t}-e^t \cdot \dfrac{1}{2\sqrt{1+e^t}}\cdot e^t}{2(1+e^t)} =\dfrac{2e^t+2e^{2t}-e^{2t}}{4(1+e^t)^\frac{3}{2}} =\dfrac{2e^t+e^{2t}}{4(1+e^t)^\frac{3}{2}} =>g''(0)=\dfrac{3}{8\sqrt{2}}$

$g(t)=\sqrt{2}+\dfrac{1}{1!\cdot 2\sqrt{2}}t+\dfrac{3}{2!\cdot 8\sqrt{2}}t^2+o(t^2)=\sqrt{2}+\dfrac{1}{2\sqrt{2}}t+\dfrac{3}{16\sqrt{2}}t^2+o(t^2)$

А тогда искомое разложение $f(x)=g(x^2)=\sqrt{2}+\dfrac{1}{2\sqrt{2}}x^2+\dfrac{3}{16\sqrt{2}}x^4+o(x^4)$

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