Problem 1
For each of the following series, find the interval and radius of convergence. a. \(\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\) b. \(\sum_{n=0}^{\infty} n ! x^{n}\) c. \(\sum_{n=0}^{\infty} \frac{(x-2)^{n}}{(n+1) 3^{n}}\)
Problem 1
In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. $$ f(x)=1+x+x^{2} \text { at } a=1 $$
Problem 1
In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial. $$ (1-x)^{1 / 3} $$
Problem 1
Suppose that the two power series \(\sum_{n=0}^{\infty} c_{n} x^{n}\) and \(\sum_{n=0}^{\infty} d_{n} x^{n}\) converge to the functions \(f\) and \(g\), respectively, on a common interval \(l\). i. The power series \(\sum_{n=0}^{\infty}\left(c_{n} x^{n} \pm d_{n} x^{n}\right)\) converges to \(f \pm g\) on \(l\). ii. For any integer \(m \geq 0\) and any real number \(b\), the power series \(\sum_{n=0}^{\infty} b x^{m} c_{n} x^{n}\) converges to \(b x^{m} f(x)\) on \(I\) iii. For any integer \(m \geq 0\) and any real number \(b\), the series \(\sum_{n=0}^{\infty} c_{n}\left(b x^{m}\right)^{n}\) converges to \(f\left(b x^{m}\right)\) for all \(x\) such that \(b x^{m}\) is in \(I\).
Problem 2
Find the interval and radius of convergence for the series \(\sum_{n=1}^{\infty} \frac{x^{n}}{\sqrt{n}}\).
Problem 2
Combining Power Series Suppose that \(\sum_{n=0}^{\infty} a_{n} x^{n}\) is a power series whose interval of convergence is \((-1,1)\), and suppose that \(\sum_{n=0}^{\infty} b_{n} x^{n}\) a power series whose interval of convergence is \((-2,2)\). a. Find the interval of convergence of the series \(\sum_{n=0}^{\infty}\left(a_{n} x^{n}+b_{n} x^{n}\right)\). b. Find the interval of convergence of the series \(\sum_{n=0}^{\infty} a_{n} 3^{n} x^{n}\).
Problem 2
In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. $$ f(x)=1+x+x^{2} \text { at } a=-1 $$
Problem 2
In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial. $$ \left(1+x^{2}\right)^{-1 / 3} $$
Problem 3
In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial. $$ (1-x)^{1.01} $$
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 \(10 \%\).
