91Ó°ÊÓ

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=2+(0.1)^{n}$$

Use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}}$$

Find a polynomial that will approximate \(F(x)\) throughout the given interval with an error of magnitude less than \(10^{-3}.\) \(F(x)=\int_{0}^{x} \tan ^{-1} t d t, \quad\) (a) \(\quad[0,0.5] \quad\) (b) \(\quad[0,1]\)

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x+2)^{n}}{n 2^{n}}$$

Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$\sum_{n=2}^{\infty}(-1)^{n+1} \frac{1}{n \ln n}$$

Use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=1}^{\infty} \frac{n(n+1)}{(n+2)(n+3)}$$

Use power series operations to find the Taylor series at \(x=0\) for the functions. $$\ln (1+x)-\ln (1-x)$$

Use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(-\ln n)^{n}}{n^{n}}$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=0}^{\infty}(-2)^{n}(n+1)(x-1)^{n}$$

Find the Taylor series generated by \(f\) at \(x=a.\) $$f(x)=1 /(1-x)^{3}, \quad a=0$$