91Ó°ÊÓ

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

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

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

(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.

True–False Determine whether the statement is true or false. Explain your answer. If \(\left\\{a_{n}\right\\}\) is eventually increasing, then \(a_{100}

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. $$ x e^{x} $$

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

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

Determine whether the statement is true or false. Explain your answer. An infinite series converges if its sequence of terms converges.

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