91Ó°ÊÓ

Find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\dots+\frac{2}{3^{n-1}}+\dots$$

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

Each of Exercises gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. $$a_{1}=1, \quad a_{n+1}=a_{n}+\left(1 / 2^{n}\right)$$

Determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{n}{10}\right)^{n}$$

Each of Exercises gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. $$a_{1}=2, \quad a_{2}=-1, \quad a_{n+2}=a_{n+1} / a_{n}$$

Find the binomial series for the functions. $$\left(1-\frac{x}{2}\right)^{4}$$

Use the limit Comparison Test to determine whether each series converges or diverges. $$\sum_{n=1}^{\infty}\left(\frac{2 n+3}{5 n+4}\right)^{n}$$

Use power series operations to find the Taylor series at \(x=0\) for the functions. $$x \ln (1+2 x)$$

Express each of the numbers as the ratio of two integers. $$0.0 \overline{6}=0.06666 \ldots$$

Find the Taylor series generated by \(f\) at \(x=a.\) $$f(x)=1 / x^{2}, \quad a=1$$