91Ó°ÊÓ

Which of the series, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{\sec ^{-1} n}{n^{1.3}}$$

A cubic approximation Use Taylor's formula with \(a=0\) and \(n=3\) to find the standard cubic approximation of \(f(x)=\) \(1 /(1-x)\) at \(x=0 .\) Give an upper bound for the magnitude of the error in the approximation when \(|x| \leq 0.1\)

Find the sum of each series. $$\sum_{n=1}^{\infty}\left(\tan ^{-1}(n)-\tan ^{-1}(n+1)\right)$$

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n}$$

Estimate the value of \(\sum_{n=1}^{\infty}\left(1 / n^{3}\right)\) to within 0.01 of its exact value.

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. \(a_{n}=\frac{\ln (n+1)}{\sqrt{n}}\)

Which series converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{2}}\right)^{n}$$

Use Taylor's formula with \(n=2\) to find the quadratic approximation of \(f(x)=(1+x)^{k}\) at \(x=0\) ( \(k\) a constant). b. If \(k=3,\) for approximately what values of \(x\) in the interval [0,1] will the error in the quadratic approximation be less

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

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