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Section 5.8

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2. (a) For -\infty < x < \infty ,

\displaystyle{\cos(x^2) = \sum_{n=0}^\infty \frac{(-1)^nx^{4n}}{(2n)!} = 1 - \frac{x^4}{2} + \frac{x^8}{4!} - \frac{x^{12}}{6!} + \frac{x^{16}}{8!} - \cdots .}

(c) For -1 < t < 1 ,

\displaystyle{\frac{1}{1 - t^2} = \sum_{n=0}^\infty t^{2n} = 1 + t^2 + t^4 + t^6 + t^8 + \cdots .}

(d) For -\dfrac{1}{2} < x < \dfrac{1}{2} ,

\displaystyle{\frac{1}{2x - 1} = -\sum_{n=1}^\infty 2^nx^n = -1 - 2x - 4x^2 - 8x^3 - 16x^4 - \cdots .}

(e) For -1 < t < 1 ,

\displaystyle{\frac{1}{(1 + t)^2} = \sum_{n=1}^\infty (-1)^{n+1}nt^{n-1} = 1 - 2t + 3t^2 - 4t^3 + 5t^4 - \cdots .}

(g) For -\infty < x < \infty ,

\displaystyle{f(x) = \sum_{n=1}^\infty \frac{(-1)^{n+1}x^{2n-1}}{(2n)!} = \frac{x}{2} - \frac{x^3}{4!} + \frac{x^5}{6!} - \frac{x^7}{8!} + \frac{x^9}{10!} - \cdots .}

3. (a) For -\infty < x < \infty ,

\displaystyle{\cos^2(x) = 1 + \sum_{n=1}^\infty \frac{(-1)^n2^{2n-1}x^{2n}}{(2n)!} = 1 - x^2 + \frac{x^4}{3} - \frac{2x^6}{45} + \frac{x^8}{315} - \cdots .}

(b) \displaystyle{P_8(x) = 1 - x^2 + \frac{x^4}{3} - \frac{2x^6}{45} + \frac{x^8}{315}}

5. (a) For -\infty < x < \infty ,

\displaystyle{\sin(x^2) = \sum_{n=0}^\infty \frac{(-1)^nx^{4n+2}}{(2n+1)!} = x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} - \frac{x^{14}}{7!} + \frac{x^{18}}{9!} - \cdots .}

(b) \displaystyle{P_{10}(x) = x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!}}

(c) For -\infty < x < \infty ,

\displaystyle{S(x) = \sum_{n=0}^\infty \frac{(-1)^nx^{4n+3}}{(4n+3)(2n+1)!} = \frac{x^3}{3} - \frac{x^7}{7(3!)} + \frac{x^{11}}{11(5!)} - \frac{x^{15}}{15(7!)} + \frac{x^{19}}{19(9!)} - \cdots .}

(d) \displaystyle{P_{11}(x) = \frac{x^3}{3} - \frac{x^7}{7(3!)} + \frac{x^{11}}{11(5!)}}

(e) S(1) \approx P_{15}(1) = 0.310268 , rounded to 6 deciimal places

7. \displaystyle{\frac{d^9}{dx^9}\mathrm{Si}(x)\Bigg|_{x=0} = \frac{1}{9}}

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