91Ó°ÊÓ

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? $$\sum_{n=0}^{\infty} x^{n}$$

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$x \ln (1+2 x)$$

If \(\cos x\) is replaced by \(1-\left(x^{2} / 2\right)\) and \(|x|<0.5,\) what estimate can be made of the error? Does \(1-\left(x^{2} / 2\right)\) tend to be too large, or too small? Give reasons for your answer.