(Solved): Find the sum of the series. \[ \sum_{n=0}^{\infty}(-1)^{n} \frac{x^{T 4 n}}{n !} \] If will be a fu ...

Find the sum of the series. \[ \sum_{n=0}^{\infty}(-1)^{n} \frac{x^{T 4 n}}{n !} \] If will be a function of the varisble \( x \). Find the Taylor series about O for each of the functions below. Give the first three non-2ero terms for each. A. \( \frac{1}{\sqrt{1+x^{1}}}=+\frac{x^{2}}{2}+\cdots \quad+\cdots \) B. \( 2 \cos (x)+x^{2}= \) For each of these series, also be sure that you can find the general term in fhe-series? Write the Taylor series for \( f(x)=e^{x} \) about \( x=-1 \) as \( \sum_{\min }^{\infty} c_{n}(x+1)^{n} \cdot \) Find the first five coelficlents. \[ \begin{array}{l} c_{4}=\square \\ c_{1}= \\ c_{2}= \\ c_{1}= \\ c_{4}= \\ \text { (7 point) } \end{array} \] WSe sigma notstien to write the Taylor series about \( z=x_{0} \) for the function. \( e^{3}+x_{0}=1 \). Tanlor senes \( =\sum_{i=5}^{\infty} \frac{1}{4}\left(x-\frac{1}{3}\right)^{h} \) Write out the first four terms of the Maclaurin series of \( f(x) \) if \[ f(0)=-10, \quad f^{\prime}(0)=-4_{i} \quad f^{\prime \prime}(0)-6_{4} \quad f^{N}(0)=11 \] \[ f(x)=+\cdots \] Approximate cos(0.2) to five decimal-place accuracy using the Mactaurin series for \( \operatorname{con}(z) \). \[ \cos (0.2)=5 \]