91Ó°ÊÓ

\(27-30\) Approximate the sum of the series correct to four decimal places. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n} n !} $$

\(27-30\) Use a power series to approximate the definite integral to six decimal places.\ $$ \int_{0}^{0.3} \frac{x^{2}}{1+x^{4}} d x $$

Find the values of \(p\) for which the series is convergent. $$\sum_{n=1}^{\infty} \frac{\ln n}{n^{p}}$$

\(21-34\) Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{k=1}^{\infty}(\cos 1)^{k}$$

Determine whether the sequence converges or diverges. If it converges, find the limit. $$ \\{\arctan 2 n\\} $$

A series \(\Sigma a_{n}\) is defined by the equations $$a_{1}=1 \quad a_{n+1}=\frac{2+\cos n}{\sqrt{n}} a_{n}$$ Determine whether \(\Sigma a_{n}\) converges or diverges.

\(3-32\) Determine whether the series converges or diverges. $$\sum_{n=1}^{\infty} \frac{n !}{n^{n}}$$

Test the series for convergence or divergence. $$\sum_{j=1}^{\infty}(-1)^{j} \frac{\sqrt{j}}{j+5}$$

Suppose you know that $$f^{(n)}(4)=\frac{(-1)^{n} n !}{3^{n}(n+1)}$$ and the Taylor series of \(f\) centered at 4 converges to \(f(x)\) for all \(x\) in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates \(f(5)\) with error less than \(0.0002 .\)

A car is moving with speed 20 \(\mathrm{m} / \mathrm{s}\) and acceleration 2 \(\mathrm{m} / \mathrm{s}^{2}\) a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute?