Chapter 1: Infinite Series, Power Series

Question: Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals.

13.$\mathbf{\sum }_{\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{n}}^{\mathbf{2}}}{{\mathbf{n}}^{\mathbf{3}}\mathbf{+}\mathbf{1}}$

A particle moves without friction under gravity on the surface of the paraboloid $\mathbf{z}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}$. Find the Lagrangian and the Lagrange equations of motion. Show that motion in a horizontal circle is possible and find the angular velocity of this motion.Use cylindrical coordinates.

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{x}}{\mathit{e}}^{\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}\mathit{d}\mathit{t}$

Question: Show that the Maclaurin series for ${(1+x)}^p$converges to${(1+x)}^p$when$0<x<1$

Give another proof of the fundamental theorem of algebra (see Problem 7.44) as follows. Let$\mathit{I}\mathbf{=}{\mathbf{\oint }}_{\left|z\right|\mathbf{=}\mathbf{R}}\frac{\mathbf{f}\mathbf{\text{'}}\mathbf{\left(}\mathbf{z}\mathbf{\right)}}{\mathbf{f}\mathbf{\left(}\mathbf{z}\mathbf{\right)}}\mathit{d}\mathit{z}$about infinity, that is, in the negative direction around a very large circle C. Use the argument principle (7.8), and also evaluate I by finding the residue of f ‘(z)/f(z) at infinity; thus show that f(z) has n zeros inside C.

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\frac{\mathbf{In}\left(1-x\right)}{\mathbf{x}}$ .

Prove that the matrix equation below using as matrix whose determinant is the Jacobian.

Find the residues at the given points.

$$\begin{gathered} a)\frac{\cos z}{{\left(2z-\pi \right)}^4}At\frac{\pi }{2} \\ b)\frac{2z^2+3z}{z-1}At\infty \\ c)\frac{z^3}{1+32z^5}Atz=\frac{1}{2} \\ d)csc\left(2z-3\right)Atz=\frac{3}{2} \end{gathered}$$

A hoop of mass $\mathbf{M}$and radius rolls without slipping down an inclined plane of angle $\mathbf{\alpha }$. Find the Lagrangian and the Lagrange equation of motion. Hint: The kinetic energy of a body which is both translating and rotating is a sum of two terms: the translational kinetic energy $\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{Mv}}^{\mathbf{2}}$where$\mathbf{v}$is the velocity of the centre of mass, and the rotational kinetic energy$\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{I\omega }}^{\mathbf{2}}$where$\mathbf{\omega }$is the angular velocity and$\mathbf{I}$is the moment of inertia around the rotation axis through the centre of mass.

Find the Maclaurin series for the following functions $\mathit{c}\mathit{o}\mathit{s}\mathbf{\left[}\mathit{l}\mathit{n}\mathbf{\right(}\mathbf{\text{1 + x}}\mathbf{\left)}\mathbf{\right]}$: