Chapter 1: Infinite Series, Power Series

Find a two-term approximation for each of the following integrals and an error bound for the given t interval.${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{t}}{\mathbf{e}}^{\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\mathbf{dx}\mathbf{,}\mathbf{0}\mathbf{<}\mathbf{t}\mathbf{<}\mathbf{0}\mathbf{.}\mathbf{1}$

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.

14.$\mathbf{\sum }_{\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{1}}{\sqrt{{\mathbf{n}}^{\mathbf{2}}\mathbf{+}\mathbf{9}}}$

A computer program gives the result$\frac{\mathbf{1}}{\mathbf{6}}$ for the sum of the series $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}{\left(-5\right)}^{\mathbf{n}}$. Show that this series is divergent. Do you see what happened? Warning hint: Always consider whether an answer is reasonable, whether it’s a computer answer or your work by hand.

Let $\mathit{A}\mathbf{=}\mathbf{2}\stackrel{\mathbf{\wedge }}{\mathbf{i}}\mathbf{-}\stackrel{\mathbf{\wedge }}{\mathbf{j}}\mathbf{+}\mathbf{2}\stackrel{\mathbf{\wedge }}{\mathbf{k}}$. (a) Find a unit vector in the same direction as $\mathit{A}$. Hint: Divide $\mathit{A}$by $\left|A\right|$. (b) Find a vector in the same direction as $\mathit{A}$but of magnitude $\mathbf{12}$. (c) Find a vector perpendicular to $\mathit{A}$. Hint: There are many such vectors; you are to find one of them. (d) Find a unit vector perpendicular to $\mathit{A}$. See hint in (a).

Find the Maclaurin series for the following functions.

$\mathbf{ln}\left(\frac{\mathrm{sinx}}{x}\right)$

Connect the midpoints of the sides of an equilateral triangle to form 4 smaller equilateral triangles. Leave the middle small triangle blank, but for each of the other 3 small triangles, draw lines connecting the midpoints of the sides to create 4 tiny triangles. Again leave each middle tiny triangle blank and draw the lines to divide the others into 4 parts. Find the infinite series for the total area left blank if this process is continued indefinitely. (Suggestion: Let the area of the original triangle be 1; then the area of the first blank triangle is 1/4.) Sum the series to find the total area left blank. Is the answer what you expect? Hint: What is the “area” of a straight line? (Comment: You have constructed a fractal called the Sierpinski gasket. A fractal has the property that a magnified view of a small part of it looks very much like the original.)

Find a two-term approximation for each of the following integrals and an error bound for the given t interval ${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{t}}\sqrt{\mathbf{X}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}\mathbf{}}\mathbf{dx}\mathbf{,}\mathbf{0}\mathbf{<}\mathbf{t}\mathbf{<}\mathbf{0}\mathbf{.}\mathbf{01}$

If you solve$(7.17)$when$f\left(x\right)=0$by assuming a solution$y=x^k$, show that the quadratic equation for$k$is the same as the auxiliary equation for the$z$equation (7.20). Thus show (see Section 5) that if the two values of$k$are equal, the second solution is not a power of$k$but is$x^{k}lnx$. Also show that if$k$is complex, say$k=a\pm bi$, the solutions are$x^{a}cos(blnx)$and$x^{a}sin(blnx)$or other equivalent forms [see$\left(5.16\right)$to$\left(5.18\right)$].

Verify that the force field is conservative. Then find a scalar potential φ such that $\mathbf{F}\mathbf{=}\mathbf{2}\mathit{x}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{y}\mathbf{i}\mathbf{-}\left(x^2+1\right)\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{y}\mathbf{j}$.

Generalize Problem $\mathbf{14}$to any mass $\mathbf{M}$of circular cross-section and moment of inertia $\mathbf{I}$. Consider a hoop, a disk, a spherical shell, a solid spherical ball; order them as to which would first reach the bottom of the inclined plane. (For moments of inertia, see Chapter $\mathbf{5}$, Section $\mathbf{4}$.)