91Ó°ÊÓ

True-False Determine whether the statement is true or false. Explain your answer. If \(\lim _{k \rightarrow+\infty}\left(u_{k+1} / u_{k}\right)=5\), then \(\sum u_{k}\) diverges.

Find the Taylor polynomials of orders \(n=0,1,2,3\), and 4 about \(x=x_{0}\), and then find the \(n\) th Taylor polynomial for the function in sigma notation. $$ \cos x ; x_{0}=\frac{\pi}{2} $$

Show that the given sequence is eventually strictly increasing or eventually strictly decreasing. $$ \left\\{\frac{n}{n^{2}+10}\right\\}_{n=1}^{+\infty} $$

(a) \(\frac{d}{d x}[\sinh x]=\cosh x\) (b) \(\frac{d}{d x}\left[\tan ^{-1} x\right]=\frac{1}{1+x^{2}}\)

Find the interval of convergence of the power series, and find a familiar function that is represented by the power series on that interval. $$ \begin{aligned} 1-(x+3)+(x+3)^{2}-&(x+3)^{3} \\ &+\cdots+(-1)^{k}(x+3)^{k} \mid \ldots \end{aligned} $$

Determine whether the series converges. $$ \sum_{k=1}^{\infty} k^{2} e^{-k^{3}} $$

Express the repeating decimal as a fraction. $$ 0.4444 \ldots $$

Prove: The Taylor series for \(\sin x\) about any value \(x=x_{0}\) converges to \(\sin x\) for all \(x\).

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

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{k=1}^{\infty} \frac{\sin k}{k^{3}} $$