91Ó°ÊÓ

Find the Maclaurin polynomials of orders \(n=0,1,2,3\) and \(4,\) and then find the \(n\) th Maclaurin polynomials for the function in sigma notation. $$x \sin x$$

Find the Maclaurin polynomials of orders \(n=0,1,2,3\) and \(4,\) and then find the \(n\) th Maclaurin polynomials for the function in sigma notation. $$x e^{x}$$

Write out the first five terms of the sequence, determine whether the sequence converges, and if so find its limit. $$\left\\{\frac{n}{2^{n}}\right\\}_{n=1}^{+\infty}$$

(a) Find an upper bound on the error that can result if \(\ln (1+x)\) is approximated by \(x\) over the interval [-0.01,0.01] (b) Check your answer in part (a) by graphing $$ |\ln (1+x)-x| $$ over the interval.

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}$$

Find the first four nonzero terms of the Maclaurin series for the function by dividing appropriate Maclaurin series. (a) \(\frac{\tan ^{-1} x}{1+x}\) (b) \(\frac{\ln (1+x)}{1-x}\)

Use the ratio test to determine whether the series converges. If the test is inconclusive, then say so. $$\sum_{k=1}^{\infty} \frac{k}{k^{2}+1}$$

Use sigma notation to write the Taylor series about \(x=x_{0}\) for the function. $$\cos x ; x_{0}=\frac{\pi}{2}$$

Determine whether the series converges. $$\sum_{k=1}^{\infty} \frac{\tan ^{-1} k}{1+k^{2}}$$

Determine whether the series converges. $$\sum_{k=1}^{\infty}\left(1+\frac{1}{k}\right)^{-k}$$