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Magazine Problem 31
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. $$h(x)=\frac{2 x^{3}}{1-x}$$
Problem 32
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\cos x, a=\pi / 6$$
Problem 32
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. $$f\left(x^{3}\right)=\frac{1}{1-x^{3}}$$
Problem 32
Power series for derivatives a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. $$f(x)=-\ln (1-x)$$
Problem 33
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\sqrt{x}, a=9$$
Problem 33
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. $$p(x)=\frac{4 x^{12}}{1-x}$$
Problem 33
Differential equations a. Find a power series for the solution of the following differential equations, subject to the given initial condition. b. Identify the function represented by the power series. $$y^{\prime}(t)-y=0, y(0)=2$$
Problem 34
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. $$f(-4 x)=\frac{1}{1+4 x}$$
Problem 34
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\sqrt[3]{x}, a=8$$
Problem 34
Differential equations a. Find a power series for the solution of the following differential equations, subject to the given initial condition. b. Identify the function represented by the power series. $$y^{\prime}(t)+4 y=8, y(0)=0$$
