Sia

“Notice that if , then and , whereas if
, then and .
(a) If is absolutely convergent, show that both of the
series and are convergent.
(b) If is conditionally convergent, show that both of the
series and are divergent.
44. Prove that if is a conditionally convergent series and
is any real number, then there is a rearrangement of
whose sum is . [Hints: Use the notation of Exercise 43.
an an

0 a
n
an
0
an an
0 an
0
an

an

an

a
n


an

an

a
n


an
r

an
r
Take just enough positive terms so that their sum is greater
than . Then add just enough negative terms so that the
cumulative sum is less than . Continue in this manner and use
Theorem 11.2.6.]
45. Suppose the series is conditionally convergent.
(a) Prove that the series is divergent.
(b) Conditional convergence of is not enough to determine whether is convergent. Show this by giving an
example of a conditionally convergent series such that
converges and an example where diverges.
r an

r

an

n
2
an

an

nan

nan

nan
an

We now have several ways of testing a series for convergence”
― James Stewart, Calculus: Early Transcendentals

“My old man also said this about Martha Knox: “She’s not beautiful, but I think she knows how to sell it.” Well, it’s true that I wanted to hold her braid. I always had wanted to from first seeing it and mostly I wanted to in that dance, but I didn’t reach for it and I didn’t set down my beer bottle. Martha Knox wasn’t selling anything. We didn’t dance again that night or again at all, because”
― Elizabeth Gilbert, Pilgrims

“Pilgrims WHEN MY OLD MAN said he’d hired her, I said, “A girl?” A girl, when it wasn’t that long ago women couldn’t work on this ranch even as cooks, because the wranglers got shot over them too much. They got shot even over the ugly cooks. Even over the old ones. I said, “A girl?” “She’s from Pennsylvania,” my old man said. “She’ll be good at this.” “She’s from what?” When my brother Crosby found out, he said, “Time for me to find new work when a girl starts doing mine.” My old man looked at him. “I heard you haven’t come over Dutch Oven Pass once this season you haven’t been asleep on your horse or reading a goddamn book. Maybe it’s time for you to find new work anyhow.” He told us that she showed up somehow from Pennsylvania in the sorriest piece of shit car he’d ever seen in his life. She asked him for five minutes to ask for a job, but it didn’t take that long. She flexed her arm for him to feel, but he didn’t feel it. He liked her, he said, right away. He trusted his eye for that, he said, after all these years. “You’ll like her, too,” he said. “She’s sexy like a horse is sexy. Nice and big. Strong.” “Eighty-five of your own horses to feed, and you still think horse is sexy,” I said, and my brother Crosby said, “I think we got enough of that kind of sexy around here already.” She was Martha Knox, nineteen years old and tall as me, thick-legged but not fat, with cowboy boots that anyone could see were new that week, the cheapest in the store and the first pair she’d ever owned. She had a big chin that worked only because her forehead and nose worked, too, and she had the kind of teeth that take over a face even when the mouth is closed. She had, most of all, a dark brown braid that hung down the center of her back, thick as a girl’s arm. I danced with Martha Knox one night early in the season. It was a day off to go down the mountain, get drunk, make phone calls, do laundry, fight. Martha Knox was no dancer. She didn’t want to dance with me. She let me know this by saying a few times that she wasn’t going to dance with me, and then, when she finally agreed, she wouldn’t let go of her cigarette. She held it in one hand and let that hand fall and not be available. So I kept my beer bottle in one hand, to balance her out, and we held each other with one arm each. She was no dancer and she didn’t want to dance with me, but we found a good slow sway anyway, each of us with an arm hanging down, like a rodeo cowboy’s right arm, like the right arm of a bull rider, not reaching for anything. She wouldn’t look anywhere but over my left shoulder, like that part of her that was a good dancer with me was some part she had not ever met and didn’t feel”
― Elizabeth Gilbert, Pilgrims

“In Depth
Types of Effect Size Indicators
Researchers use several different statistics to indicate effect size depending on the nature of their data. Roughly
speaking, these effect size statistics fall into three broad categories. Some effect size indices, sometimes called dbased effect sizes, are based on the size of the difference between the means of two groups, such as the difference between the average scores of men and women on some measure or the differences in the average scores
that participants obtained in two experimental conditions. The larger the difference between the means, relative
to the total variability of the data, the stronger the effect and the larger the effect size statistic.
The r-based effect size indices are based on the size of the correlation between two variables. The larger the
correlation, the more strongly two variables are related and the more of the total variance in one variable is systematic variance related to the other variable.
A third category of effect sizes index involves the odds-ratio, which tells us the ratio of the odds of an
event occurring in one group to the odds of the event occurring in another group. If the event is equally likely in
both groups, the odds ratio is 1.0. An odds ratio greater than 1.0 shows that the odds of the event is greater in
one group than in another, and the larger the odds ratio, the stronger the effect. The odds ratio is used when the
variable being measured has only two levels. For example, imagine doing research in which first-year students in
college are either assigned to attend a special course on how to study or not assigned to attend the study skills
course, and we wish to know whether the course reduces the likelihood that students will drop out of college.
We could use the odds ratio to see how much of an effect the course had on the odds of students dropping out.
You do not need to understand the statistical differences among these effect size indices, but you will
find it useful in reading journal articles to know what some of the most commonly used effect sizes are called.
These are all ways of expressing how strongly variables are related to one another—that is, the effect size.
Symbol Name
d Cohen’s d
g Hedge’s g
h
2 eta squared
v
2
omega squared
r or r
2 correlation effect size
OR odds ratio
The strength of the relationships between
variables varies a great deal across studies. In some
studies, as little as 1% of the total variance may be
systematic variance, whereas in other contexts,
the proportion of the total variance that is systematic
variance may be quite large,”
― Mark R. Leary, Introduction to Behavioral Research Methods