Chapter 1: Infinite Series, Power Series

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.

$\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\frac{\mathbf{tanx}\mathbf{-}\mathbf{x}}{{\mathbf{x}}^{\mathbf{3}}}$.

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 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 $\left(r,\theta \right)$, 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.

$\mathbf{\sum }_{\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{n}}{{\left(n^2+1\right)}^{\mathbf{2}}}$

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

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

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

If you invest a dollar at “$\mathbf{\text{6\%}}$interest compounded monthly,” it amounts to ${\left(1.005\right)}^{\mathbf{n}}$dollars after months. If you invest $\mathbf{\$}\mathbf{10}$at the beginning of each month for 10 years (120 months), how much will you have at the end of the 10 years?