What is a $p$ -series?

The $p$ -Series Test

10-51.

BALANCING BOOKS

The diagram at right shows books hanging over the edge of a desk. In each case, the books have been placed so that the total overhang is as large as possible.

Continuing the pattern in the diagram, what is the total overhang for five books?
Write a series for the total overhang of a stack of an infinite number of books using summation notation.
Does this series converge or diverge? Give evidence for your answer. What does this tell you about how far a stack with an infinite number of books may extend past the edge of a desk?

Top with 1 book overhanging 1 half, middle with 2 books overhanging 1 half & 1 fourth, bottom with 3 books overhanging 1 half & 1 fourth & 1 sixth. Your teacher will provide you with a model.

10-52.

A series of the form $S = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { p } }$ is referred to as a $\boldsymbol p$ -series. Some $p$ -series converge while others do not.

Write out the first four terms of series $S$ for the values of $p$ below. Then use the Integral Test to test each series for convergence.

$p = 0.9$

$p = l$

$p = 2$

10-53.

The $p$ -Series Test

Using your results from problem 10-52, copy and complete the statement below to write a conjecture about the convergence of $p$ -series.
The $\boldsymbol p$ -Series Test
Given an infinite series of the form $\displaystyle \sum _{n=k} ^ \infty \frac{1}{n^p}$ , where $k>0$ ,
if ____________, then the series _____________.
Write the inverse of your $p$ -Series Test conjecture. Is it always true as well? If so, add the inverse to the conjecture in part (a). If not, give a counterexample.

10-54.

Create two different series $S = \displaystyle\sum _ { n = 1 } ^ { \infty } a _ { n }$ and $T = \displaystyle\sum _ { n = 1 } ^ { \infty } b _ { n }$ $\left(b_{n} ≠ 0\right)$ such that $S$ and $T$ both diverge yet the series $U = \displaystyle\sum _ { n = 1 } ^ { \infty } ( a _ { n } - b _ { n } )$ converges.

10-55.

Revisit the list of infinite series in problem 10-2. Identify which series can be conveniently tested with the $p$ -Series Test. Then, for each series, use the $p$ -Series Test to determine if the series converges.

10-56.

Solve for $y$ if $\frac { d y } { d x } = \frac { x y } { \operatorname { ln } ( y ) }$ . Homework Help ✎

10-57.

Examine the following series. Use one of the tests you have learned so far to determine if each series converges or diverges. Name the tests that you used. Homework Help ✎

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \operatorname { ln } ( n ) }$

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

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

$\displaystyle\sum _ { n = 1 } ^ { \infty } 2 ( \frac { 5 } { 4 } ) ^ { n }$

10-58.

Express $\ln(4.5)$ in terms of $\ln(3)$ and $\ln(2)$ . Use any of the properties of logarithms listed below. Homework Help ✎

$\ln(ab) = \ln(a) + \ln(b)$

$\ln(\frac { a } { b }) = \ln(a) - \ln(b)$

$\ln(a^b) = b \ln(a)$

10-59.

As Miranda drives her car, her acceleration is $a(t) = te^{2t}$ . What is her average acceleration over $0 ≤ t ≤ 2$ seconds? Homework Help ✎

10-60.

What is $\frac { d y } { d x }$ for each of the following equations? Homework Help ✎

$6xy - \cos^2(x) = \sqrt { 2 y }$

$y^{\prime\prime} =\frac { 2 } { x ^ { 2 } - 3 x }$

$y = \int _ { 2 } ^ { x } 5 t ^ { 3 } d t$

$y = \ln(\ln(\ln x))$

10-61.

A solid has a circular base of radius $r = 2$ units. If the cross-sections perpendicular to the base are square, calculate the volume of the solid. 10-61 HW eTool (Desmos). Homework Help ✎

10-62.

Without graphing, determine the maximum value of $y = x^3 - 3x^2 + 2x + 4$ over $0 ≤ x ≤ 3$ . Homework Help ✎

Calculus Third Edition