91Ó°ÊÓ

Find the first four nonzero terms of the Maclaurin series for the function by dividing appropriate Maclaurin series. $$ \frac{x^{3}+x^{2}+2 x-2}{x^{2}-1} $$

Write out the first five terms of the sequence, determine whether the sequence converges, and if so find its limit. $$ \left\\{\left(\frac{n+3}{n+1}\right)^{n}\right\\}_{n=1}^{+\infty} $$

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

Confirm the derivative formula by differentiating the appropriate Maclaurin series term by term. (a) \(\frac{d}{d x}[\cos x]=-\sin x\) (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. $$ 1+(x-2)+(x-2)^{2}+\cdots+(x-2)^{k}+\cdots $$

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

Express the repeating decimal as a fraction. $$ 0.9999 \ldots $$

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

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

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