91Ó°ÊÓ

. Find the third-order Maclaurin polynomial for \((1+x)^{1 / 2}\) and bound the error \(R_{3}(x)\) for \(-0.5 \leq x \leq 0.5\).

Find the third-order Maclaurin polynomial for \((1+x)^{3 / 2}\) and bound the error \(R_{3}(x)\) if \(-0.1 \leq x \leq 0\).

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems 43-48 to see that you get the same answers using the methods of Section 9.7. $$ \exp (\sin x) $$

Show that it is possible for \(\Sigma a_{n}\) and \(\Sigma b_{n}\) both to diverge and yet for \(\Sigma\left(a_{n}+b_{n}\right)\) to converge.

Test for convergence or divergence. (a) \(\sum_{n=1}^{\infty} \sin ^{2}\left(\frac{1}{n}\right)\) (b) \(\sum_{n=1}^{\infty} \tan \left(\frac{1}{n}\right)\) (c) \(\sum_{n=1}^{\infty} \sqrt{n}\left[1-\cos \left(\frac{1}{n}\right)\right]\)

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems 43-48 to see that you get the same answers using the methods of Section 9.7. $$ (\sin x)(\exp x) $$

Find the third-order Maclaurin polynomial for $$ (1+x)^{-1 / 2} $$ Ind bound the error \(R_{3}(x)\) if \(-0.05 \leq x \leq 0.05\).

Let \(S=\left\\{x: x\right.\) is rational and \(\left.x^{2}<2\right\\}\). Convince yourself that \(S\) does not have a least upper bound in the rational numbers, but does have such a bound in the real numbers. In other words, the sequence of rational numbers \(1,1.4,1.41,1.414, \ldots\), has no limit within the rational numbers.

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems 43-48 to see that you get the same answers using the methods of Section 9.7. $$ (\sin x) /(\exp x) $$

Find the fourth-order Maclaurin polynomial for $$ \ln [(1+x) /(1-x)] $$ and bound the error \(R_{4}(x)\) for \(-0.5 \leq x \leq 0.5\).