91Ó°ÊÓ

Determine convergence or divergence of the series. $$\sum_{k=1}^{\infty} \frac{1}{\cos ^{2} k}$$

Determine the radius and interval of convergence. $$\sum_{k=2}^{\infty} k^{2}(x-3)^{k}$$

Einstein's theory of relativity states that the mass of an object traveling at velocity \(v\) is \(m(v)=m_{0} / \sqrt{1-v^{2} / c^{2}},\) where \(m_{0}\) is the rest mass of the object and \(c\) is the speed of light. Show that \(m \approx m_{0}+\left(\frac{m_{0}}{2 c^{2}}\right) v^{2} .\) Use this approximation to estimate how large \(v\) would need to be to increase the mass by \(10 \%\)

Use graphical and numerical evidence to conjecture the convergence or divergence of the series. $$\sum_{k=1}^{\infty} \frac{1}{k^{2}}$$

Determine whether the series is convergent or divergent. $$\sum_{k=0}^{\infty}(-1)^{k+1} \frac{1}{2^{k}}$$

Determine whether the sequence converges or diverges. $$a_{n}=\frac{n 2^{n}}{3^{n}}$$

Determine whether the series is absolutely convergent, conditionally convergent or divergent. $$\sum_{k=1}^{\infty} \frac{\cos k}{k^{3}}$$

Determine the radius and interval of convergence. $$\sum_{k=4}^{\infty} \frac{1}{k^{2}}(x+2)^{k}$$

Prove that the Taylor series converges to \(f(x)\) by showing that \(R_{n}(x) \rightarrow 0\) as \(n \rightarrow \infty\). $$e^{-x}=\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{k}}{k !}$$

Use graphical and numerical evidence to conjecture the convergence or divergence of the series. $$\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}$$