91Ó°ÊÓ

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\tan ^{-1} n $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1}{\sqrt{n}} \tan ^{-1} n $$

The accompanying figure shows the first three rows and part of the fourth row of a sequence of rows of semicircles. There are \(2^{n}\) semicircles in the \(n\) th row, each of radius \(1 / 2^{n} .\) Find the sum of the areas of all the semicircles.

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(\frac{1}{3}\right)^{n}+\frac{1}{\sqrt{2^{n}}} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\sqrt[n]{n^{2}+n} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{(\ln n)^{200}}{n} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{(\ln n)^{5}}{\sqrt{n}} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=n-\sqrt{n^{2}-n} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1}{\sqrt{n^{2}-1}-\sqrt{n^{2}+n}} $$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(23-84\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1}{n} \int_{1}^{n} \frac{1}{x} d x $$