91Ó°ÊÓ

Find the first four terms of the binomial series for the functions. \begin{equation}(1-2 x)^{1 / 2}\end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=\ln (1+x), \quad a=0\)

In Exercises \(1-8,\) use the Direct Comparison Test to determine if each series converges or diverges. $$\sum_{n=2}^{\infty} \frac{n+2}{n^{2}-n}$$

Use the Integral Test to determine if the series in Exercises \(1-12\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{1}{n+4} $$

In Exercises \(1-36,\) (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely (c) conditionally? $$ \sum_{n=1}^{\infty} \frac{(3 x-2)^{n}}{n} $$

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=\frac{2^{n}}{2^{n+1}} $$

Use the Integral Test to determine if the series in Exercises \(1-12\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} e^{-2 n} $$

In Exercises \(1-6,\) find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\cdots+\frac{1}{(n+1)(n+2)}+\cdots$$

In Exercises \(1-36,\) (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely (c) conditionally? $$ \sum_{n=0}^{\infty} \frac{(x-2)^{n}}{10^{n}} $$

In Exercises \(1-8,\) use the Ratio Test to determine if each series converges absolutely or diverges. $$ \sum_{n=1}^{\infty} \frac{n^{4}}{(-4)^{n}} $$