Problem 3
The following exercises consider problems of annuity payments. For annuities with a present value of \(\$ 1\) million, calculate the annual payouts given over 25 years assuming interest rates of \(1 \%, 5 \%\), and \(10 \%\).
Problem 3
In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. $$ f(x)=\cos (2 x) \text { at } a=\pi $$
Problem 3
Suppose that \(\sum_{n=0}^{\infty} a_{n} x^{n}\) has an interval of convergence of \((-1,1)\). Find the interval of convergence of \(\sum_{n=0}^{\infty} a_{n}\left(\frac{x}{2}\right)^{n}\).
Problem 4
In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial. $$ (1-2 x)^{2 / 3} $$
Problem 4
Sketch a graph of \(f(x)=\frac{1}{1-x^{2}}\) and the corresponding partial sums \(S_{N}(x)=\sum_{n=0}^{N} x^{2 n}\) fo \(N=2,4,6\) on the interval \((-1,1)\)
Problem 4
In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. $$ f(x)=\sin (2 x) \text { at } a=\frac{\pi}{2} $$
Problem 5
In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. $$ f(x)=\sqrt{x} \text { at } a=4 $$
Problem 5
Use a power series to represent each of the following functions \(f\). Find the interval of convergence. a. \(f(x)=\frac{1}{1+x^{3}}\) b. \(f(x)=\frac{x^{2}}{4-x^{2}}\)
Problem 5
Use the series for \(f(x)=\frac{1}{1-x}\) on \(|x|<1\) to construct a series for \(\frac{1}{(1-x)(x-2)} .\) Determine the interval of convergence.
Problem 6
Represent the function \(f(x)=\frac{x^{3}}{2-x}\) using a power series and find the interval of convergence.
