Problem 97
In the following exercises, find the Taylor series at the given value. $$ f(x)=\frac{3}{x}, a=1 $$
Problem 98
In the following exercises, find the Maclaurin series for the given function. $$ f(x)=e^{-x^{2}}-1 $$
Problem 99
In the following exercises, find the Maclaurin series for the given function. $$ f(x)=\cos x-x \sin x $$
Problem 100
In the following exercises, find the Maclaurin series for \(F(x)=\int_{0}^{x} f(t) d t\) by integrating the Maclaurin series of \(f(x)\) term by term. $$ f(x)=\frac{\sin x}{x} $$
Problem 101
In the following exercises, find the Maclaurin series for \(F(x)=\int_{0}^{x} f(t) d t\) by integrating the Maclaurin series of \(f(x)\) term by term. $$ f(x)=1-e^{x} $$
Problem 102
In the following exercises, find the Maclaurin series for \(F(x)=\int_{0}^{x} f(t) d t\) by integrating the Maclaurin series of \(f(x)\) term by term. Use power series to prove Euler's formula: \(e^{i x}=\cos x+i \sin x\)
Problem 104
The following exercises consider problems of annuity payments. A lottery winner has an annuity that has a present value of \(\$ 10\) million. What interest rate would they need to live on perpetual annual payments of \(\$ 250,000 ?\)
Problem 105
The following exercises consider problems of annuity payments. Calculate the necessary present value of an annuity in order to support annual payouts of \(\$ 15,000\) given over 25 years assuming interest rates of \(1 \%, 5 \%\), and \(10 \%\).
