91Ó°ÊÓ

Find the binomial series for the functions in Exercises \(11-14\) . $$ (1+x)^{4} $$

Write out the first few terms of each series to show how the series starts. Then find the sum of the series. $$ \sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right) $$

Find the Maclaurin series for the functions in Exercises \(9-20 .\) $$ \frac{1}{1+x} $$

Which of the series in Exercises \(11-44\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1}(0.1)^{n} $$

Each of Exercises \(7-12\) 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. $$ a_{1}=a_{2}=1, \quad a_{n+2}=a_{n+1}+a_{n} $$

Which of the series in Exercises 1–36 converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{(\ln n)^{2}}{n^{3}} $$

Find the Maclaurin series for the functions in Exercises \(9-20 .\) $$ \frac{1}{1-x} $$

Find the binomial series for the functions in Exercises \(11-14\) . $$ \left(1+x^{2}\right)^{3} $$

Find Taylor series at \(x=0\) for the functions in Exercises \(7-18\) $$ x^{2} \cos \left(x^{2}\right) $$

Each of Exercises \(7-12\) 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. $$ a_{1}=2, \quad a_{2}=-1, \quad a_{n+2}=a_{n+1} / a_{n} $$