Chapter 1: Infinite Series, Power Series

: Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection or by$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{18}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{23}\mathbf{)}\mathbf{,}$or.$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{24}\mathbf{)}\mathbf{,}$Also find a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [see$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{26}\mathbf{)}\mathbf{,}$and the discussionafter$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{15}\mathbf{)}\mathbf{,}$].

$\mathbf{(}\mathbf{4}{\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbf{D}\mathbf{+}\mathbf{5}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{40}{\mathit{e}}^{\mathbf{-}\mathbf{3}\mathbf{x}\mathbf{/}\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{x}$

Use the ratio test to find whether the following series converge or diverge:

19.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{3}}^{\mathbf{n}}}{{\mathbf{2}}^{\mathbf{2}\mathbf{n}}}$

In $\mathbf{(}\mathbf{x}\mathbf{+}\sqrt{\mathbf{1}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}}\mathbf{)}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{x}}\frac{\mathbf{dt}}{\sqrt{\mathbf{1}\mathbf{+}{\mathbf{t}}^{\mathbf{2}}}}$

a. Show that it is possible to stack a pile of identical books so that the top book is as far as you like to the right of the bottom book. Start at the top and each time place the pile already completed on top of another book so that the pile is just at the point of tipping. (In practice, of course, you can't let them overhang quite this much without having the stack topple. Try it with a deck of cards.) Find the distance from the right-hand end of each book to the right-hand end of the one beneath it. To find a general formula for this distance, consider the three forces acting on book, and write the equation for the torque about its right-hand end. Show that the sum of these setbacks is a divergent series (proportional to the harmonic series).

b. By computer, find the sum ofterms of the harmonic series with.

$\mathit{N}\mathbf{=}\mathbf{25}\mathbf{,}\mathbf{100}\mathbf{,}\mathbf{200}\mathbf{,}\mathbf{1000}\mathbf{,}{\mathbf{10}}^{\mathbf{6}}\mathbf{,}{\mathbf{10}}^{\mathbf{100}}$

c. From the diagram in (a), you can that with 5 books (count down from the top) the top book is completely to the right of the bottom book, that is, the overhang is slightly over one book. Use your series in (a) to verify this. Then using parts (a) and (b) and a computer as needed, find the number of books needed for an overhang of 2 books, 3 books, 10 books, or 100 books.

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don’t forget the preliminary test. Use the facts stated above when they apply

$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{3}\mathbf{)}}$

Change the independent variable to simplify the Euler equation, and then find a first integral of it.

$\mathbf{1}\mathbf{.}\mathbf{}\mathbf{}{\mathbf{\int }}_{{\mathbf{x}}_{\mathbf{1}}}^{{\mathbf{x}}_{\mathbf{2}}}{\mathbf{y}}^{\raisebox{1ex}{$\mathbf{3}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.}\mathbf{ds}$

In the bouncing ball example above, find the height of the tenth rebound, and the distance traveled by the ball after it touches the ground the tenth time. Compare this distance with the total distance traveled.

Prove theorem $\mathbf{14}\mathbf{.}\mathbf{3}$. Hint: Group the terms in the error as role="math" localid="1657423688910" $\mathbf{(}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{2}}\mathbf{)}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{3}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{4}}\mathbf{)}\mathbf{+}\mathbf{\cdots }$to show that the error has the same sign as role="math" localid="1657423950271" ${\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{.}$Then group them asrole="math" localid="1657423791335" ${\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{2}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{3}}\mathbf{)}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{4}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{5}}\mathbf{)}\mathbf{+}\mathbf{\cdots }}$to show that the error has magnitude less than$\mathbf{\left|}{\mathit{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{\right|}\mathbf{.}$

First simplify each of the following numbers to the $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$form or to the$\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

$\frac{\mathbf{1}}{\mathbf{1}\mathbf{+}\mathbf{i}}$.

Question:Find the steady-state temperature distribution inside a sphere of radius 1 when the surface temperatures are as given in Problems 1 to 10.
$\mathbf{35}{\mathbf{cos}}^{\mathbf{4}}\mathbf{\theta }$.