91Ó°ÊÓ

Give the first four terms of the sequences for which \(a_{n}\) is given. $$a_{n}=n^{2}, \quad n=1,2,3, \ldots$$

Find at least three nonzero terms (including \(a_{0}\) and at least two cosine terms and two sine terms if they are not all zero) of the Fourier series for the given functions, and sketch at least three periods of the function. $$f(x)=\left\\{\begin{array}{rr}1 & -\pi \leq x<0 \\\0 & 0 \leq x<\pi\end{array}\right.$$

Calculate the value of each of the given functions. Use the indicated number of terms of the appropriate series. Compare with the value found directly on a calculator. $$e^{0.2}\quad(3)$$

Find the first three nonzero terms of the Maclaurin expansion of the given functions. $$f(x)=e^{x}$$

Give the first four terms of the sequences for which \(a_{n}\) is given. $$a_{n}=\frac{2^{n+1}}{n !}, n=1,2,3, \ldots$$

Find the first three nonzero terms of the Maclaurin expansion of the given functions. $$f(x)=\sin x$$

Find at least three nonzero terms (including \(a_{0}\) and at least two cosine terms and two sine terms if they are not all zero) of the Fourier series for the given functions, and sketch at least three periods of the function. $$f(x)=\left\\{\begin{array}{ll}0 & -\pi \leq x<-\frac{\pi}{2}, \frac{\pi}{2} \leq x<\pi \\\2 & -\frac{\pi}{2} \leq x<\frac{\pi}{2}\end{array}\right.$$

Give the first four terms of the sequences for which \(a_{n}\) is given. $$a_{n}=\frac{1}{n+2}, \quad n=0,1,2, \ldots$$

Determine whether the given function is even, or odd, or neither. One period is defined for each function.$$f(x)=\left\\{\begin{array}{lr}5 & -3 \leq x<0 \\\0 & 0 \leq x<3\end{array}\right.$$.

Find at least three nonzero terms (including \(a_{0}\) and at least two cosine terms and two sine terms if they are not all zero) of the Fourier series for the given functions, and sketch at least three periods of the function. $$f(x)=\left\\{\begin{array}{ll}0 & -\pi \leq x<-\frac{\pi}{2}, \frac{\pi}{2} \leq x<\pi \\\2 & -\frac{\pi}{2} \leq x<\frac{\pi}{2}\end{array}\right.$$