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. $$ \sin \pi x $$

Determine whether the series converges, and if so find its sum. $$ \sum_{k=2}^{\infty} \frac{1}{k^{2}-1} $$

Determine whether the series converges. $$ \sum_{k=1}^{\infty} \frac{3}{5 k} $$

Use the limit comparison test to determine whether the series converges. \(\sum_{k=1}^{\infty} \frac{1}{(2 k+3)^{17}}\)

Use the ratio \(a_{n+1} / a_{n}\) to show that the given sequence \(\left\\{a_{n}\right\\}\) is strictly increasing or strictly decreasing. $$ \left\\{\frac{10^{n}}{(2 n) !}\right\\}_{n=1}^{+\infty} $$

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. $$ \ln (1+x) $$

Use the ratio test for absolute convergence (Theorem 9.6.5) to determine whether the series converges or diverges. If the test is inconclusive, say so. $$ \sum_{k=1}^{\infty}(-1)^{k} \frac{k^{3}}{e^{k}} $$

Use sigma notation to write the Taylor series about \(x=x_{0}\) for the function. $$ e^{x} ; x_{0}=1 $$

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

Determine whether the series converges. $$ \sum_{k=1}^{\infty} \frac{1}{\sqrt{k+5}} $$