91Ó°ÊÓ

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

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

Is it true that if \(\sum_{n=1}^{\infty} a_{n}\) is a divergent series of positive numbers, then there is also a divergent series \(\Sigma_{n=1}^{\infty} b_{n}\) of positive numbers with \(b_{n}

Use Theorem 20 to find the series' interval of convergence and, within this interval, the sum of the series as a function of \(x\) $$\sum_{n=0}^{\infty}\left(\frac{\sqrt{x}}{2}-1\right)^{n}$$

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

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

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 \(x=0\) $$f(x)=\sin x$$

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 \(x=0\) $$f(x)=\tan x$$

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

Linearizations at inflection points Show that if the graph of a twice- differentiable function \(f(x)\) has an inflection point at \(x=a,\) then the linearization of \(f\) at \(x=a\) is also the quadratic approximation of \(f\) at \(x=a\). This explains why tangent lines fit so well at inflection points.