10.3.1I've got the power!

Power Series Convergence

10-127.

So far, you have investigated series of terms that are real numbers. In the next few problems you will be introduced to polynomial series that have variables in the terms. Their utility will be introduced now, and explored further in Chapter 12.

Write out the first four terms of $S = \displaystyle \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k }$ and $T = \displaystyle\sum _ { k = 1 } ^ { \infty } \frac { x ^ { k } } { k }$ .
Note that $T = S$ when $x = 1$ . Write out four terms of $T$ when $x = 2$ and when $x = −1$ .
Is $S$ convergent or divergent? Justify your answer.
Is $T$ convergent when $x = 2$ ? When $x = −1$ ? Justify your answer.
Using the Ratio Test, between what values of $x$ will $T$ be a convergent series?
Finally, determine if the interval in your answer for part (e) should be an open or closed interval. In other words, should the endpoints be included in the interval of convergence?

Hint: Substitute each endpoint, one at a time, into series $T$ , and choose an appropriate test to determine if the new series converges or diverges.

Math Notes

Power Series

Given a variable $x$ , then the infinite series of the form

$\displaystyle\sum _ { k = 1 } ^ { \infty } a _ { k } x ^ { k } = a _ { 0 } + a _ { 1 } x +$ $a _ { 2 } x ^ { 2 } + \ldots + a _ { n } x ^ { n } +\ldots$

is called a power series, because it is a series of powers of $x$ .

We may also have a power series centered at $c$ , if it is of the form:

$\displaystyle \sum _ { k = 1 } ^ { \infty } a _ { k } ( x - c ) ^ { k } = a _ { 0 } +$ $a _ { 1 } ( x - c ) + a _ { 2 } ( x - c ) ^ { 2 } + \ldots +$ $a _ { n } ( x - c ) ^ { n } \ldots$

The convergence of a power series depends on the value of $x$ . Every power series $\sum a _ { n } ( x - c ) ^ { n }$ has a radius of convergence $R$ (non-negative), such that the series converges if $| x - c | < R$ and diverges if $| x - c | > R$ . The series may or may not converge where $| x - c | = R$ . The set of all $x$ -values where the series converges is the interval of convergence.

Example: If a power series converges on the interval $[5, 11)$ , then $R = 3$ because $\frac { 11 - 5 } { 2 } = 3$ .

10-128.

What is the difference between a $p$ -series and power series?

10-129.

Determine the radius of convergence for each power series below.

$\displaystyle \sum _ { n = 2 } ^ { \infty } ( x - 2 ) ^ { n }$

$\displaystyle\sum _ { n = 2 } ^ { \infty } ( n - 2 ) ! x ^ { n }$

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

10-130.

To determine the interval and radius of convergence of a power series, you can use any of the tests you have learned so far. Then, solve for $x$ so that the series will converge. Determine the radius and interval of convergence for each power series below.

$S = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { x ^ { n } } { n }$

$T = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { x ^ { n } } { n ! }$

$U = \displaystyle\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x + 1 ) ^ { n } } { 2 ^ { n } }$

$V = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 5 ^ { n } } { n ^ { 2 } } x ^ { n }$

10-131.

Determine if each of the following series converges or diverges. State the tests you used. Homework Help ✎

$\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 ^ { n } - 1 }$

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

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } + n } { n ! + 2 }$

10-132.

Evaluate each of the following limits. Homework Help ✎

$\lim\limits _ { x \rightarrow 4 } \frac { \sqrt { x } - 2 } { x - 4 }$

$\lim\limits _ { x \rightarrow 0 } \frac { 2 ^ { x } - 3 ^ { x } } { x ^ { 2 } }$

$\lim\limits _ { x \rightarrow 1 ^ { + } } \frac { \operatorname { ln } ( x ) } { \sqrt { x - 1 } }$

$\lim\limits _ { x \rightarrow 0 ^ { + } } \frac { \operatorname { ln } ( x ) } { e ^ { 1 / x } }$

$\lim\limits _ { x \rightarrow \pi / 2 } \frac { \operatorname { sec } ( x ) } { \operatorname { sec } ( 2 x ) }$

10-133.

The velocity of a particle is given by $v(t) = t \cos(π t)$ . Homework Help ✎

For what value(s) of $t$ is the particle at rest?
What is the acceleration of the particle at $t = 2$ ?
Write a function $x(t)$ for the position if $x(0) = 5$ .

10-134.

Determine the dimensions of the trapezoid of greatest area inscribed under the curve $y = 9 -x^2$ if one base is along the $x$ -axis. 10-134 HW eTool (Desmos). Homework Help ✎

10-135.

Demonstrate that the function $F ( x ) = \int _ { x } ^ { 4 x } \frac { 1 } { t } d t$ is constant for all positive values of $x$ . Homework Help ✎

10-136.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int \operatorname { sec } ^ { 4 } ( x ) \operatorname { tan } ^ { 3 } ( x ) d x$

$\int x \operatorname { tan } ^ { - 1 } ( x ) d x$

$\int \frac { \operatorname { ln } ( x ) } { x } d x$

$\int \frac { \operatorname { cos } \sqrt { x } } { \sqrt { x } } d x$

10-137.

Multiple Choice: If $x^2y = 4$ , then $\frac { d y } { d x } =$ Homework Help ✎

$\frac { x ^ { 2 } } { y }$

$\frac { 2 } { \sqrt { x } }$

$\frac { - y } { 2 x }$

$\frac { - 2 y } { x }$

$4 \sqrt { x } y$

10-138.

Multiple Choice: A function that satisfies the differential equation $f^\prime(x) + f(x) = 0$ is: Homework Help ✎

$f(x) = \sin (x)$

$f(x) = \cos (x)$

$f(x) = e^x$

$f(x) = e^{−x}$

$f(x) = \ln (x)$

Calculus Third Edition

Power Series