91Ó°ÊÓ

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

In Exercises \(1-32,\) (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=0}^{\infty} \frac{3^{n} x^{n}}{n !} $$

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

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

In Exercises \(13-22,\) find a formula for the \(n\) th term of the sequence. The sequence \(1,-1,1,-1,1, \ldots\)

Find Taylor series at \(x=0\) for the functions in Exercises \(7-18\) $$ \cos ^{2} x\left(\text {Hint} : \cos ^{2} x=(1+\cos 2 x) / 2 .\right) $$

Converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.) $$ \sum_{n=0}^{\infty} \frac{-2}{n+1} $$

Which of the series in Exercises \(1-26\) converge, and which diverge? Give reasons for your answers. (When checking your answers, remember there may be more than one way to determine a series' convergence or divergence.) $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right) $$