Problem 15
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n !}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot(2 n-1)} $$
Problem 16
In Exercises \(7-28,\) find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} n !(x-4)^{n}}{3^{n}} $$
Problem 16
Use the Integral Test to determine the convergence or divergence of the series, where \(k\) is a positive integer. $$ \sum_{n=1}^{\infty} n^{k} e^{-n} $$
Problem 16
Use the binomial series to find the Maclaurin series for the function. $$ f(x)=\frac{1}{\sqrt{1-x}} $$
Problem 16
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2(-1)^{n+1}}{e^{n}+e^{-n}}=\sum_{n=1}^{\infty}(-1)^{n+1} \operatorname{sech} n $$
Problem 16
Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(\frac{7}{2}, 4, \frac{9}{2}, 5, \ldots\)
Problem 16
Use the power series $$\frac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^{n} x^{n}$$ to determine a power series, centered at 0 , for the function. Identify the interval of convergence. $$ h(x)=\frac{x}{x^{2}-1}=\frac{1}{2(1+x)}-\frac{1}{2(1-x)} $$
Problem 16
In Exercises \(7-18\), find the Maclaurin polynomial of degree \(n\) for the function. $$ f(x)=\frac{x}{x+1}, \quad n=4 $$
Problem 17
In Exercises \(7-18\), find the Maclaurin polynomial of degree \(n\) for the function. $$ f(x)=\sec x, \quad n=2 $$
Problem 17
In Exercises \(17-20\), approximate the sum of the series by using the first six terms. (See Example 4.) $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} 3}{n^{2}} $$
