sum_(n=2)^oo lnn/n^p Find the positive values of p for which the series converges.

To find the convergence of the series sum_(n=2)^oo (ln(n))/n^p where pgt0 (positive values of p ), we may apply integral test.
Integral test is applicable if f is positive, continuous, and decreasing function on an interval and let a_n=f(x). Then the infinite series sum_(n=k)^oo a_n converges if and only if the improper integral int_k^oo f(x) dx converges to a real number. If the integral diverges then the series also diverges.
For the infinte series series sum_(n=2)^oo (ln(n))/n^p , we have:
a_n =(ln(n))/n^p
Then, f(x) =(ln(x))/x^p
The f(x) satisfies the conditions for integral test when pgt0 . We set-up the improper integral as:
int_2^oo (ln(x))/x^pdx
Apply integration by parts: int u dv = uv - int v du.
Let: u=ln(x) then du = 1/xdx
       dv = 1/x^p dx
Then , v = int dv
              =int 1/x^p dx
              = int x^(-p) dx
             = x^(-p+1)/(-p+1)
The indefinite integral will be:
int (ln(x))/x^pdx = ln(x)x^(-p+1)/(-p+1)- intx^(-p+1)/(-p+1) *1/x dx
                    = ln(x)x^(-p+1)/(-p+1)-1/(-p+1) int (x^(-p)x)/x dx
                   = ln(x)x^(-p+1)/(-p+1)-1/(-p+1) intx^(-p) dx        
                  = ln(x)x^(-p+1)/(-p+1)-1/(-p+1) *x^(-p+1)/(-p+1)
                   =(ln(x)x^(-p+1))/(-p+1)-x^(-p+1)/(-p+1)^2
                  =(ln(x)x^(-p+1))/(-p+1)*(-p+1)/(-p+1)-x^(-p+1)/(-p+1)^2
                  =(ln(x)x^(-p+1)(-p+1))/(-p+1)^2-x^(-p+1)/(-p+1)^2
                 =(ln(x)x^(-p+1)(-p+1)-x^(-p+1))/(-p+1)^2|_2^oo
The definite integral will only be finite if 1-p<0 or pgt1 .
Thus, the series  sum_(n=2)^oo(ln(n))/n^p converges when pgt1 .

Why does Scout want to stop playing the game?

The "Boo Radley" game begins in chapter 4 of To Kill a Mockingbird. The game involves the children playing various roles, with the part of Boo Radley belonging to Jem. Their drama is "woven from bits and scraps of gossip and neighborhood legend."
Scout wants to quit the game for two reasons. One day, Atticus happens to see the children playing the game. Atticus asks, "Does this have anything to do with the Radleys?" Jem denies that it does. This is the first reason Scout wants to quit the game. She doesn't want to be in trouble with Atticus.
The second reason Scout wants to quit the game involves a previous incident. Earlier, when Scout, Jem, and Dill are playing outside, Jem rolls Scout in a tire. She lands in the Radley's front yard after bumping into the steps of their porch. She doesn't share this information with the others, but when she falls out of the tire, she believes that "Someone was inside the house laughing." This is her second reason she thinks it would be best to stop playing the "Boo Radley" game. She is fairly certain that Boo must have seen them.

Calculus: Early Transcendentals, Chapter 4, Review, Section Review, Problem 33

y=(x-2)e^(-x)
(I) Asymptotes
To determine its horizontal asymptotes, take the limit of this function as x approaches positive and negative infinity.
lim_(x->-oo) (x-2)e^(-x) =-oo
lim_(x->oo)(x-2)e^(-x)=0
So, the function has only one horizontal asymptote. Its horizontal asymptote is y=0 .
Moreover, there are no values of x which make the function undefined. Thus, the function has no vertical asymptotes.
(II) Maximum/Minimum Point
To determine the maximum or minimum point of a function, take its derivative.
y=(x-2)e^(-x)
y'=(x-2)*e^(-x)*(-1) + e^(-x)*(1)
y'= e^(-x) ((-1)(x-2)+1)
y'=e^(-x)(-x + 3)
Set the derivative equal to zero.
0 = e^(-x) (-x+ 3)
To solve for x, we only have to consider the factor (-x + 3). Since the first factor e^(-x) will never be zero for any value of x.
0=-x + 3
x=3
So, there is only one critical number, which is x=3 .
To determine if the function is maximum or minimum at x=3, refer to the change of signs of first derivative before and after the critical number. If the change of signs of y' is from negative to positive, then, the function is minimum at that critical number. If the change of signs of y' is from positive to negative, then, the function is maximum at that point.
At the left of x=3, the interval is (-oo, 3) . And at the right of it, the interval is (3,oo) . Take a test value for each interval and plug-in them to the first derivative.
y' = e^(-x)(-x + 3)
For the interval (-oo,3) , let the test value be x=0.
y'=e^(0)(0+3) = 3
For the interval (3,oo) , let the test value be x=4.
y'=e^(-4)(-4+3)=e^(-4)(-1) = -0.018
Since the change of signs of y' is from positive to negative, then, the function is maximum at x=3. The value of y when x=3 is:
y=(x-2)e^(-x)
y=(3-2)e^(-3)=0.05
Hence, the function has a maximum point (3, 0.05). And it has no minimum point.
(III) Increasing/Decreasing Interval
To determine at which interval is the function increasing and decreasing, refer to the sign of the first derivative. If the sign of the first derivative is positive, the function is increasing at that interval. If the sign is negative, the function is decreasing.
At the left of the critical number, the interval is (-oo,3) . In this interval, the value of y' is positive.
And at the right of the critical number, the interval is (3,oo) . The value of y' here is negative.
Thus, the function is increasing in the interval (-oo,3) and it is decreasing in the interval (3,oo) .
(IV) Inflection point
To determine the inflection point, take the second derivative of the function.
y'=e^(-x)(-x +3)
y'' = e^(-x)*(-1) + (-x+3)*e^(-x)*(-1)
y''=e^(-x)(-1 + (-x+3)(-1))
y''=e^(-x)(x-4)
Set the second derivative equal to zero.
0=e^(-x)(x-4)
To solve for the value of x, consider only the factor (x-4). It is because the factor e^(-x) will never be zero for any value of x.
0=x-4
4=x
So the function changes concavity at x=4 only. The value of y when x=4 is:
y=(x-2)e^(-x)
y=(4-2)e^(-4) = 2e^(-4)=0.04
Thus, the inflection point is (4, 0.04).
(V) Concavity
To determine the concavity of the function, consider the region at the left and right of x=4.
At the left of x=4, the interval (-oo,4) . And at the right of it, the interval is (4,oo) .
Take a test value for each interval and plug-in them to the second derivative.
y''=e^(-x)(x-4)
If the second derivative is negative, the function is concave downward in that interval. If the value of the second derivative is positive, the function is concave upward.
For the first interval (-oo,4) , let the test value be x=0.
y''=e^0(0-4)=-4 (Concave down)
For the second interval (4,oo), let the test value be x=5.
y''=e^(-5)(5-4)=0.007 (Concave up)
Thus, the function is concave down in the interval (-oo,4) and it is concave up in the interval (4,oo) .
(VI) Graph
Therefore, the graph of the function y = (x-2)e^(-x) is:

What are the themes in the story "The Lottery" by Shirley Jackson?

Without question, tradition is a huge theme within "The Lottery," as the very existence of tradition is expressed in such a brutal way. It would be hard to read the story and not wonder about the ridiculous notion of carrying on a tradition where, for no reason, someone has to die, yet there the community is, carrying on the tradition, regardless.
Another theme at work though has to do with selfish interest since the vast majority of the outspoken cries against the lottery are only provided once the family has been chosen. Since the chosen understandably don't want to die, the time for speaking out has arrived, but it's drenched in that selfish interest. Their outcries against the lottery would have felt more solid and unbiased if they would've come before the event happened, which could be a reason why so little happens in consequence to those outcries. From there then, we're stepping into a theme of timing as well. 
Yet another theme would be the prettying up of matters that are unpleasant, perhaps society's tendency to do so. The community holds the lottery in a very organized way, and it has this ironic name that has a connotation of winning and gain. These details are things that make the situation feel less barbaric, though there can be no doubt that the circumstance is, in fact, barbaric.

A couple of major themes of this story have to do with the subject of tradition. First, the story conveys that people do not like to go against tradition, even if they do not particularly care for the tradition itself. The narrator says that when Mr. Summers spoke to the town about making a new black box for use during the lottery, people responded poorly because "no one liked to upset even as much tradition as was represented by the black box." Despite the fact that it's literally just a painted wooden box, and a box that has splintered and cracked and faded over the years, people are resistant to replacing it because it's the only box they really remember ever using. This notion seems pretty ridiculous: they are so stuck in their ways that they cannot see the value in replacing a "shabby" wooden box.
Second, the story conveys the theme that traditions should continually be evaluated for their cultural relevance and humanity. Clearly, this tradition of choosing one person to stone every month is inhumane and does nothing to better the community. There's talk, we learn, of another town over discontinuing their lottery, and so we know that such a reevaluation can take place. We surmise by the cruel way Tessie Hutchinson is stoned to death by her friends and family that such a reevaluation would be positive and right.

How does Grover react when Percy tells him about the yarn-cutting?

Percy witnesses some old ladies at a fruit stand bring out a pair of scissors, large and apparently symbolic, having previously watched them knitting a pair of what seemed to be enormous, electric-blue socks. He then sees them cut the yarn.
Later, Grover asks Percy what he saw at the fruit stand, and Percy, puzzled by Grover's strange behavior, says that he saw the middle of the three old ladies bring out a pair of scissors and cut the yarn. When he hears this, Grover closes his eyes and makes a gesture which Percy interprets as akin to crossing himself, but not quite the same thing. Nervously, Grover begins chewing at his thumb and says he doesn't want things to be "like last time." He mentions that "they never get past sixth" and asks to walk Percy home. He seems very concerned that something terrible is about to happen.
Percy asks whether this is "a superstition thing" or if Grover thinks somebody is going to die. Grover does not respond in words but "looked at me mournfully, like he was already picking the kind of flowers I'd like best on my coffin."
After this, disturbed by Grover's behavior, Percy leaves him at the bus station when Grover goes to the bathroom.

How can behavior be seen as a result of heredity or the environment?

The truth of the matter is that who we are is the result of both heredity and the environment. It is not "nature versus nurture," but a combination of nature and nurture. I also want to point out that "behavior" does not seem to be a very precise term to use in this context, as behavior is simply what we do, which is not necessarily consistent over a lifetime or even over a day. Behavior is situational to a very large degree and seldom consistent enough to try to analyze it in this way. To the degree that this is the exact issue you are expected to address, behavior is environmental because behavior cannot be inherited at all. Our personalities, on the other hand, seem to be fairly consistent over our lives, and it is these that are a combination of nature and nurture.
Some traits, like shyness, are tendencies some people are born with—a genetic predisposition. If that predisposition is reinforced by one's environment, consistently, one is likely to become a shy adult. If that predisposition is not reinforced and a child is provided with opportunities to be more outgoing, he or she may become more outgoing with time. An inhibited child born into a rowdy family may retreat into even greater inhibition. A child who is gently exposed to the opportunity to be gradually more outgoing and uninhibited with very small birthday parties or family get-togethers is unlikely to be an extremely outgoing adult, but is not going to have to hide in the bathroom at a large event.
We are all born with various predispositions that can be enhanced or repressed by our environments. Our behaviors, on the other hand, are situational, as behavior is what we do, and there is nothing hereditary about taking a walk, attending a class, or washing the dishes.
https://www.psychologytoday.com/us/blog/breaking-the-ice/200806/are-we-born-shy

Please explain and analyze the poem "Barbie Doll" by Marge Piercy.

Marge Piercey's poem "Barbie Doll" is an indictment of the socially constructed values of beauty which are forced upon women from the time that they are young.
In the first stanza of the poem, we are introduced to the female character, the "girlchild," who is provided with stereotypically "feminine" toys which imply her inherited responsibilities as a future mother ("dolls that did pee-pee"), homemaker ("miniature GE stoves and irons"), and debutante presented for the visual consumption of others ("wee lipsticks the color of cherry candy").
While the girl is able to survive this childhood and live into puberty, she is soon cut down by the cruelty of a classmate, who comments on her changing body: "You have a great big nose and fat legs."
This is a deeply ironic moment; although the girl has been pushed toward womanhood for the entirety of her life, she is greeted with objectification and disgust the very moment she arrives there. Despite her many wonderful qualities--her health, intelligence, strength, and appetites--all those who are framing her as an object can see are those physical features which they deem unsuitable. 
In the third stanza, the girl is now subjected to the advice of those who wish to shape her into a more easily digested being--a "beautiful" woman. They urge her to "exercise, diet, smile and wheedle," and thereby invert all of her natural qualities. The exhaustion of this playacting wears the girl down until she finally resorts to drastic measures: cutting off her nose and legs to please those around her. 
This, of course, results in the death of the girl. Yet, as she lays in her casket, painted with makeup by the undertaker, she has finally achieved what the rest of the world has desired for her: beauty. It was only in destroying her personhood--her life--that she was able to obtain the approval of others.
This is a disgusting reflection of the kind of subversive thinking that is socialized in young women; girls are brought up to believe that their sole value lies in their appearance and ability to perform as a "feminine" woman. Thus, the last two lines of the poem ("Consummation at last. / To every woman a happy ending.") are sarcastic ones, said with the knowledge that the social systems constructed around women are ones that profit off the death of their dignity.