91Ó°ÊÓ

Use the \(n\)th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=1}^{\infty} \frac{n}{n+10}$$

Find the Taylor series generated by \(f\) at \(x=a\). $$f(x)=1 / x^{2}, \quad a=1$$

Use power series operations to find the Taylor series at \(x=0\) for the functions. $$\frac{x}{3} \ln \left(1+x^{2}\right)$$

Which of the series, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=3}^{\infty} \frac{1}{\ln (\ln n)}$$

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

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty}(-1)^{n} n^{2}(2 / 3)^{n}$$

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. $$\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}}$$

Which of the series, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{(\ln n)^{2}}{n^{3}}$$

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