91Ó°ÊÓ

${\mathbf{∫}}_{\mathbf{0}}^{\mathbf{x}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{d}\mathit{t}$

Show as in Problem 11that the Maclaurin series for${\mathit{e}}^{\mathbf{x}}$converges to${\mathit{e}}^{\mathbf{x}}$.

In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if$\mathit{n}\mathbf{=}\mathbf{2}$, the water can be made as pure as you like, but that$\mathit{n}\mathbf{=}\mathbf{2}$ if, at least one-half of the impurity will remain no matter how many stages are used.

Find out whether infinity is a regular point, an essential singularity, or a pole (and if a pole, of what order) for each of the following functions. Find the residue of each function at infinity, $\mathbf{ln}\frac{\mathbf{z}\mathbf{+}\mathbf{1}}{\mathbf{z}\mathbf{-}\mathbf{1}}$

Find the Lagrangian and Lagrange's equations for a simple pendulum (Problem $\mathbf{4}$) if the cord is replaced by a spring with spring constant $\mathbf{k}$. Hint: If the unstretched spring length is ${\mathbf{r}}_{\mathbf{8}}$, and the polar coordinates of the mass $\mathbf{m}$are $\left(r,\mathrm{θ}\right)$, the potential energy of the spring is $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{k}{\left(r-r_0\right)}^{\mathbf{2}}$.

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

$\underset{\mathbf{1}}{\overset{\mathbf{∞}}{\mathbf{∑}}}\frac{\mathbf{n}}{{\left(n^2+1\right)}^{\mathbf{2}}}$

Verify that the sets listed in (13.7e) are groups. Hint: See the proofs in (13.7f).

Show that, in a group multiplication table, each element appears exactly once in each row and in each column. Hint: Suppose that an element appears twice, and show that this leads to a contradiction, namely that two elements assumed different are the same element.

Find the interval of convergence, including end-point tests:$\underset{\mathbf{n}\mathbf{=}\mathbf{1}}{\overset{\mathbf{∞}}{\mathbf{∑}}}\frac{{\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{)}}^{\mathbf{n}}}{{\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{)}}^{\mathbf{n}}\sqrt{\mathbf{n}}}$

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