91Ó°ÊÓ

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

Find the sum of each series. $$\sum_{n=1}^{\infty} \frac{40 n}{(2 n-1)^{2}(2 n+1)^{2}}$$

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. \(a_{n}=\sin \left(\frac{\pi}{2}+\frac{1}{n}\right)\)

Use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(2 n) !}$$

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

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

Use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(2 n+3)\left(2^{n}+3\right)}{3^{n}+2}$$

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

Find the sum of each series. $$\sum_{n=1}^{\infty} \frac{2 n+1}{n^{2}(n+1)^{2}}$$

The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a .\) In Exercises \(41-46,\) find the (a) linearization (Taylor polynomial of order 1 ) and (b) quadratic approximation of \(f\) at \(x=0\) $$f(x)=\cosh x$$