91Ó°ÊÓ

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

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

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

a. 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 than \(1 / 100 ?\)

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

Recursively Defined Terms Which of the series \(\Sigma_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=5, \quad a_{n+1}=\frac{\sqrt[n]{n}}{2} a_{n}$$

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

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{10^{n}}$$

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

Use a geometric series to represent each of the given functions as a power series about \(x=0,\) and find their intervals of convergence. a. \(f(x)=\frac{5}{3-x}\) b. \(\quad g(x)=\frac{3}{x-2}\)