91Ó°ÊÓ

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$\sum_{k=1}^{\infty} \sin \frac{k \pi}{2}$$

Determine whether the series converges. $$\sum_{k=1}^{\infty} k^{2} \sin ^{2}\left(\frac{1}{k}\right)$$

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. $$\sin \pi x ; x_{0}=\frac{1}{2}$$

Show that the given sequence is eventually strictly increasing or eventually strictly decreasing. $$\left\\{2 n^{2}-7 n\right\\}_{n=1}^{+\infty}$$

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

Confirm the derivative formula by differentiating the appropriate Maclaurin series term by term. (a) \(\frac{d}{d x}[\sinh x]=\cosh x\) (b) \(\frac{d}{d x}\left[\tan ^{-1} x\right]=\frac{1}{1+x^{2}}\)

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

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

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.

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