91Ó°ÊÓ

Use the geometric series $$f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k}, \quad \text { for }|x|<1$$ to find the power series representation for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f\left(x^{3}\right)=\frac{1}{1-x^{3}}$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\cos x, a=\pi / 6$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)-y(t)=0, y(0)=2$$

Use the geometric series $$f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k}, \quad \text { for }|x|<1$$ to find the power series representation for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$p(x)=\frac{4 x^{12}}{1-x}$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\sqrt{x}, a=9$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)+4 y(t)=8, y(0)=0$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\sqrt[3]{x}, a=8$$

Use the geometric series $$f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k}, \quad \text { for }|x|<1$$ to find the power series representation for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f(-4 x)=\frac{1}{1+4 x}$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f(3 x)=\ln (1-3 x)$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)-3 y(t)=10, y(0)=2$$