10.1.1Does it converge?

Convergence of Series

10-1.

Explain how the diagram at right helps to justify that the infinite series below converges. Then, determine the sum.

$\frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 } + \frac { 1 } { 32 } +...$

Your teacher will provide you with a model.

10-2.

Examine the different infinite series presented below. Your goal is to decide which series converge (have a finite sum). Rewrite each series using sigma notation. As you work through each example, try to figure out what feature of the series is most critical in its convergence or divergence.

$1 + \frac { 3 } { 2 } + \frac { 9 } { 4 } + \frac { 27 } { 8 } + \ldots$

$\frac { 1 } { 2 } + \frac { 1 } { 5 } + \frac { 1 } { 10 } + \frac { 1 } { 17 } + \frac { 1 } { 26 } + \ldots$

$1 - 2 + 3 - 4 + ...$

$\frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \frac { 1 } { 20 } + \ldots$

$- 2 + 1 - \frac { 2 } { 3 } + \frac { 1 } { 2 } - \frac { 2 } { 5 } + \frac { 1 } { 3 } \dots$

$\operatorname { ln } ( \frac { 1 } { 2 } ) + \operatorname { ln } ( \frac { 2 } { 3 } ) + \operatorname { ln } ( \frac { 3 } { 4 } ) + . . .$

$100 - 90 + 81 - 72.9 + ...$

$1 + \frac { 1 } { 8 } + \frac { 1 } { 27 } +\frac { 1 } { 64 } + .. .$

$1 + \frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 24 } + \frac { 1 } { 120 } + . . .$

$\frac { 1 } { 2 } + 1 + \frac { 9 } { 8 } + 1 + \frac { 25 } { 32 } + \frac { 36 } { 64 } + \ldots$

$\frac { 1 } { 2 } + \frac { 2 } { 3 } + \frac { 3 } { 4 } + \frac { 4 } { 5 } + \frac { 5 } { 6 } . . .$

$1 + \frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { 3 } } + \frac { 1 } { 2 } + \frac { 1 } { \sqrt { 5 } } + \ldots$

10-3.

Your teacher will assign your team one or two of the series from problem 10-2 for a presentation. Consolidate your ideas to share with the class. Be sure to include:

The interesting qualities of this series.
If the series converges or not. Why or why not?
How you can tell if this series converges or not.

10-4.

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 ✎

$\int x ^ { 2 } e ^ { - x } d x$

$\int x \operatorname { sec } ( x ^ { 2 } ) d x$

$\int _ { - \infty } ^ { \infty } \frac { 1 } { 1 + x ^ { 2 } } d x$

$\int _ { a } ^ { b } f ^ { \prime } ( x ) d x$

10-5.

Sketch a graph and estimate the area that lies outside $r = 2 + 2\cos(3θ)$ and inside $r = 4$ . 10-5 HW eTool (Desmos). Homework Help ✎

10-6.

The table below shows the rate of books being checked out (in books per hour) from the local library at various times (in hours) during the day. Homework Help ✎

Time	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
Rate	$16$	$39$	$47$	$68$	$81$	$77$	$85$	$62$

About how many books were checked out during the day?
What was the average rate of books being checked out?

10-7.

Examine the graph of $y = f(t)$ shown at right. Homework Help ✎

Let $F ( x ) = \int _ { 0 } ^ { 2 x - 4 } f ( t ) d t$ .

What is the domain of $F$ ?
For what values of $x$ does $F$ attain its absolute maximum and minimum? Justify your answer.

Continuous linear piecewise labeled, f of x, starting at the point (0, comma 1), turning right at the point (2, comma negative 1), & ending at the point (6, comma negative 1).

10-8.

Multiple Choice: The solution to the differential equation $\frac { d y } { d x } = 6 x y$ if $(0, 4)$ lies on the graph of the solution curve is: Homework Help ✎

$y =4 e ^ { 3 x ^ { 2 } }$

$y =e ^ { 3 x ^ { 2 } }$

$y =e ^ { 3 x ^ { 2 } }$

$y = 3x^2 + 4$

none of these

10-9.

Multiple Choice: The equation of the line tangent to the curve $3x^2y - 2x = y^2$ at the point $(1, 1)$ is: Homework Help ✎

$y = -4x + 1$

$y = -4x + 5$

$y =\frac { 1 } { 2 }x + 1$

$y = \frac { 1 } { 2 }x+\frac { 1 } { 2 }$

none of these

10-10.

Multiple Choice: If $f ( x ) = \left\{ \begin{array} { l l } { 3 x ^ { 3 / 2 } + 2 } & { \text { for } x \leq 4 } \\ { 6 \sqrt { 2 x + 8 } + 1 } & { \text { for } x > 4 } \end{array} \right.$ , then $\lim\limits _ { x \rightarrow 4 } f ( x ) =$ Homework Help ✎

$24$

$25$

$26$

10-11.

Multiple Choice: No calculator! A point of inflection for the curve $y = \frac { 12 } { x ^ { 2 } + 4 }$ is: Homework Help ✎

$( - \frac { 2 \sqrt { 3 } } { 3 } , \frac { 9 } { 4 } )$

$( \frac { 2 \sqrt { 3 } } { 3 } , - \frac { 9 } { 4 } )$

$(0, 3)$

$( \sqrt { 2 } , 2 )$

$( - \sqrt { 2 } , 2 )$

Compute without a calculator

Calculus Third Edition