91Ó°ÊÓ

Finding the Radius of Convergence In Exercises \(5-10\) , find the radius of convergence of the power series. $$ \sum_{n=0}^{\infty}(-1)^{n} \frac{x^{n}}{n+1} $$

In Exercises 5–16, find a power series for the function, centered at and determine the interval of convergence. $$ f(x)=\frac{1}{3-x}, \quad c=1 $$

Finding a First-Degree Polynomial Approximation In Exercises \(5-8,\) find a first-degree polynomial function \(P_{1}\) whose value and slope agree with the value and slope of \(f\) at \(x=c .\) Use a graphing utility to graph \(f\) and \(P_{1} .\) What is \(P_{1}\) called? $$ f(x)=\frac{\sqrt{x}}{4}, \quad c=4 $$

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=(-1)^{n+1}\left(\frac{2}{n}\right) $$

Using the Direct Comparison Test In Exercises \(3-12\) , use the Direct Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=2}^{\infty} \frac{1}{\sqrt{n}-1} $$

Finding a Taylor Series In Exercises \(1-12,\) use the definition of Taylor series to find the Taylor series, centered at \(c,\) for the function. $$ f(x)=\frac{1}{x}, \quad c=1 $$

Determine the convergence or divergence of the series. $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n+1}$$

Using the Integral Test In Exercises \(1-22,\) confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} e^{-n} $$

Finding a Taylor Series In Exercises \(1-12,\) use the definition of Taylor series to find the Taylor series, centered at \(c,\) for the function. $$ f(x)=\frac{1}{1-x}, \quad c=2 $$

In Exercises 1–6, write the first five terms of the sequence. $$ a_{n}=2+\frac{2}{n}-\frac{1}{n^{2}} $$