91Ó°ÊÓ

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{n}=\frac{n+1}{2 n-1}\)

In Exercises 1-12, use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{2 n^{2}+1}\)

In Exercises \(1-34\), determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}}\)

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n+2} $$

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=0}^{\infty} \frac{x^{n}}{n+1} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{1}{n^{4}} $$

In Exercises \(1-10\), use Equation (1) to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=e^{2 x}, \quad c=0\)

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{n}=\frac{(-1)^{n+1} 2^{n}}{n+1}\)

Use Equation (1) to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=e^{-3 x}, \quad c=0\)

Use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+2 n}\)