91Ó°ÊÓ

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. $$ \text { (a) } \frac{d}{d x}[\cos x]=-\sin x \quad \text { (b) } \frac{d}{d x}[\ln (1+x)]=\frac{1}{1+x} $$

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

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

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

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

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. $$ \cos x ; \quad x_{0}=\frac{\pi}{2} $$

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.

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

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