Problem 12
In Exercises \(7-18\), find the Maclaurin polynomial of degree \(n\) for the function. $$ f(x)=\sin \pi x, \quad n=3 $$
Problem 13
Verify that the infinite series diverges. $$ \sum_{n=1}^{\infty} \frac{2^{n}+1}{2^{n+1}} $$
Problem 13
Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2 n}{n^{2}+1} $$
Problem 13
In Exercises \(7-18\), find the Maclaurin polynomial of degree \(n\) for the function. $$ f(x)=x e^{x}, \quad n=4 $$
Problem 13
Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{2}{1-x^{2}}, \quad c=0 $$
Problem 13
Prove that the Maclaurin series for the function converges to the function for all \(x\). $$ f(x)=\sinh x $$
Problem 13
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \sqrt{n}}{n+2} $$
Problem 14
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.) $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n}}{(n+1)(n+2)} $$
Problem 14
Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n}{n^{4}+1} $$
Problem 14
Prove that the Maclaurin series for the function converges to the function for all \(x\). $$ f(x)=\cosh x $$
