91Ó°ÊÓ

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 \(x=0\) $$f(x)=\ln (\cos x)$$

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

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. \(a_{n}=\frac{1}{(0.9)^{n}}\)

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

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

Find the sum of each series. $$\sum_{n=1}^{\infty} \frac{6}{(2 n-1)(2 n+1)}$$

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 \(x=0\) $$f(x)=e^{\sin x}$$

Which of the series, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=2}^{\infty} \frac{1}{n !}$$ (Hint: First show that \((1 / n !) \leq(1 / n(n-1)) \text { for } n \geq 2 .)\)

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 \(x=0\) $$f(x)=1 / \sqrt{1-x^{2}}$$

Use the identity \(\sin ^{2} x=(1-\cos 2 x) / 2\) to obtain the Maclaurin series for \(\sin ^{2} x\). Then differentiate this series to obtain the Maclaurin series for \(2 \sin x \cos x .\) Check that this is the series for \(\sin 2 x\).