Review Exercises

True or False? In the following exercises, justify your answer with a proof or a counterexample.

If the radius of convergence for a power series $\sum_{n = 0}^{\infty} a_{n} x^{n}$ is $5,$ then the radius of convergence for the series $\sum_{n = 1}^{\infty} n a_{n} x^{n - 1}$ is also $5 .$

254.

Power series can be used to show that the derivative of $e^{x} is e^{x} .$ (Hint: Recall that $e^{x} = \sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} .)$

255.

For small values of $x, sin x \approx x .$

256.

The radius of convergence for the Maclaurin series of $f (x) = 3^{x}$ is $3 .$

In the following exercises, find the radius of convergence and the interval of convergence for the given series.

257.

$\sum_{n = 0}^{\infty} n^{2} {(x - 1)}^{n}$

258.

$\sum_{n = 0}^{\infty} \frac{x^{n}}{n^{n}}$

259.

$\sum_{n = 0}^{\infty} \frac{3 n x^{n}}{12^{n}}$

260.

$\sum_{n = 0}^{\infty} \frac{2^{n}}{e^{n}} {(x - e)}^{n}$

In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.

261.

$f (x) = \frac{x^{2}}{x + 3}$

262.

$f (x) = \frac{8 x + 2}{2 x^{2} - 3 x + 1}$

In the following exercises, find the power series for the given function using term-by-term differentiation or integration.

263.

$f (x) = {tan}^{−1} (2 x)$

264.

$f (x) = \frac{x}{{(2 + x^{2})}^{2}}$

In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?

265.

$f (x) = x^{3} - 2 x^{2} + 4, a = −3$

266.

$f (x) = e^{1 / (4 x)}, a = 4$

In the following exercises, find the Maclaurin series for the given function.

267.

$f (x) = cos (3 x)$

268.

$f (x) = ln (x + 1)$

In the following exercises, find the Taylor series at the given value.

269.

$f (x) = sin x, a = \frac{π}{2}$

270.

$f (x) = \frac{3}{x}, a = 1$

In the following exercises, find the Maclaurin series for the given function.

271.

$f (x) = e^{− x^{2}} - 1$

272.

$f (x) = cos x - x sin x$

In the following exercises, find the Maclaurin series for $F (x) = \int_{0}^{x} f (t) d t$ by integrating the Maclaurin series of $f (x)$ term by term.

273.

$f (x) = \frac{sin x}{x}$

274.

$f (x) = 1 - e^{x}$

275.

Use power series to prove Euler’s formula: $e^{i x} = cos x + i sin x$

The following exercises consider problems of annuity payments.

276.

For annuities with a present value of $$ 1$ million, calculate the annual payouts given over $25$ years assuming interest rates of $1 %, 5 %, and 10 % .$

277.

A lottery winner has an annuity that has a present value of $$ 10$ million. What interest rate would they need to live on perpetual annual payments of $$ 250,000 ?$

278.

Calculate the necessary present value of an annuity in order to support annual payouts of $$ 15,000$ given over $25$ years assuming interest rates of $1 %, 5 %, and 10 % .$