Chapter 1: Infinite Series, Power Series

In Problems 5 to 7, use Fermat’s principle to find the path followed by a light ray if the index of refraction is proportional to the given function.

${\mathbf{x}}^{\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}}$

$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{(}\mathbf{-}\mathbf{1}{\mathbf{)}}^{\mathbf{n}}{\mathbf{x}}^{\mathbf{n}}}{\mathbf{\left(}\mathbf{2}\mathbf{n}\mathbf{\right)}\mathbf{!}}$

Show thatis the distance between the points and in the complex plane. Use this result to identify the graphs in Problems $\mathbf{53}\mathbf{,}\mathbf{54}\mathbf{,}\mathbf{59}\mathbf{,}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathbf{60}$without computation.

Test for convergence:$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{2}}^{\mathbf{\infty }}\frac{\sqrt{\mathbf{n}\mathbf{-}\mathbf{1}}}{{\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}$

There are 9 one-digit numbers (1 to 9), 90 two-digit numbers (10 to 99). How many three-digit, four-digit, etc., numbers are there? The first 9 terms of the harmonic series $\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{+}\mathbf{ }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\frac{\mathbf{1}}{\mathbf{9}}$are all greater than $\frac{\mathbf{1}}{\mathbf{10}}$; similarly consider the next 90 terms, and so on. Thus prove the divergence of the harmonic series by comparison with the series

$$\begin{gathered} \left[\frac{1}{10}+\frac{1}{10}+... \left(9 terms each=\frac{1}{10}\right)+\left[90 terms each=\frac{1}{100}\right]\right]\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.} \\ \mathbf{=}\frac{\mathbf{9}}{\mathbf{10}}\mathbf{+}\frac{\mathbf{90}}{\mathbf{100}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{=}\frac{\mathbf{9}}{\mathbf{10}}\mathbf{+}\frac{\mathbf{9}}{\mathbf{10}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.} \end{gathered}$$

Write and solve the Euler equations to make the following integrals stationary. In solving the Euler equations, the integrals in Chapter 5, Section 1, may be useful.

${\mathbf{\int }}_{{\mathbf{x}}_{\mathbf{1}}}^{{\mathbf{x}}_{\mathbf{2}}}\left(y{\text{'}}^2+\sqrt{y}\right)\mathit{d}\mathit{x}$

A particle moves on the surface of a sphere of radius ‘$\mathbf{a}$’ under the action of the earth’s gravitational field. Find the $\mathbf{\theta }\mathbf{,}\mathbf{\phi }$equations of motion. (Comment: This is called a spherical pendulum. It is like a simple pendulum suspended from the center of the sphere, except that the motion is not restricted to a plane.)

Find the work done by the force is $\mathit{F}\mathbf{=}\left(2xy-3\right)\mathit{i}\mathbf{+}{\mathit{x}}^{\mathbf{2}}\mathit{j}$ in moving an object from (1,0) to (0,1) long each of the three paths shown:

(a) straight line,

(b) circular arc,

(c) along lines parallel to the axes.

Given ,$\mathbf{z}\mathbf{=}{\left(x+y\right)}^{\mathbf{5}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{=}\mathbf{sin}\mathbf{10}\mathbf{x}\mathbf{}\mathbf{,}\mathbf{}\mathbf{find}\mathbf{}\frac{\mathbf{dz}}{\mathbf{dx}}\mathbf{.}$.

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.

6.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{!}}$