How was a criminal executed in brodingnag?

On Brobdingnag, Gulliver receives the dubious honor of being invited to an execution. Gulliver's normally averse to such grisly spectacles, but his curiosity gets the better of him and he agrees to go along. The criminal in question is a murderer, and as is customary on Brobdingnag, he is to be beheaded for his wicked crime. The murderer is sat down in a large chair upon a scaffold, where he will subsequently be decapitated. The execution itself is mercifully swift, the murderer's head being removed with one quick, single blow from a sword forty feet in length.
After the deed is done, a huge spurt of blood and assorted viscera shoots into the atmosphere like a great fountain at Versailles. The executed murderer's head tumbles down onto the scaffold floor, causing the ground to shake beneath Gulliver's tiny feet, even though he's standing at least half a mile away.

How can I compare the ways in which Animal Farm and The Crucible condemn the misuse of power?

The two novels present the misuse of power in different ways, but the situations have some commonalities.
In The Crucible, the government of the town remains intact, but everyone in it is tainted—including Reverend Hale, an outsider. The powers that be aim to return the town to its supposed pre-crisis harmonious normalcy. The threat of witchcraft in the town is ostensibly spiritual but turns out to be social.
In Animal Farm, some of the community members stage a revolution. They aim to create a radically different form of government, an allegorical representation of communism. Corruption builds gradually as a few individuals crave power and manipulate others to get it.
Shared features include characters' use of manipulation and deceit as a way to get what they want. Sacrifice for the greater good, and the futility of it, are also common in both. Blindness to the excesses wrought by corruption is another commonality, as the characters forge ahead on a misguided path despite evidence of imminent disaster.

One way of comparing the two is by looking at how the respective authority figures fall short of living up to their principles. In Animal Farm, Napoleon claims to have established a new society based on the principles of Animalism. Having been liberated from the cruel tyranny of Mr. Jones, the animals are supposedly now free to lead happy, fulfilling, prosperous lives. The reality, of course, is very different. Far from creating paradise on earth, Napoleon and his gang have created a brutal tyranny in which the animals are terrorized, starved, and forced to slave away in atrocious conditions, largely for the benefit of the ruling pigs.
In The Crucible, Salem is effectively a Calvinist theocracy. The people of the town pride themselves on being God-fearing Christian folk. Yet there is precious little evidence of Christ's teachings in the hysterical craze that holds the town in a vicelike grip. There is not much in the way of loving one's neighbor in the endless round of false accusations and lies that swirl about the town like dead leaves in the fall.
In both Animal Farm and The Crucible, tyrannies have been established in places where they really ought not to have been.

Throughout Animal Farm, Orwell condemns the misuse of power by depicting how Napoleon tyrannically rules the farm. Napoleon requires the animals to work long hours, barely feeds them, and suppresses their individual freedoms using threats, manipulation, and propaganda. Napoleon selfishly uses power to advance his own agenda and secure his position as leader. The other animals suffer because of his tyrannical decisions and some animals even lose their lives during Napoleon's violent purges. Napoleon is always quick to punish those who challenge his authority and conditions on the farm mirror those under Mr. Jones's reign by the end of the novella. Throughout the play The Crucible, Arthur Miller depicts how the officials of Salem refuses to recognize that their court is corrupt. Deputy Governor Danforth and Judge Hawthorne reject any criticism of their rulings and believe that any person challenging their authority is colluding with the Devil. Despite ample evidence that their court is corrupt, Danforth uses his position of authority to advance his agenda. As a result, honorable men and women die because of the court officials' stubborn, malevolent decisions. Both works depict how authority figures suppress individual rights and use their positions to selfishly advance their own agendas. Innocent citizens suffer at the hands of oppressive, selfish authority figures throughout both works.

Can you explain what the zone is in Roadside Picnic? Give an example from the reading using specific details.

The Zone is an alien landing site. The citizens of Earth wake up one day to learn that aliens have temporarily stopped by on Earth and have left several spots of close contact, which get named “Zones.” These zones correspond to ships landing at random on a spinning globe when launched from a great distance.
The Zone has many interesting properties and is filled with exotic matter, unique forms of radiation, and groundbreaking technology that was discarded by the aliens upon their departure. The event is likened to a group of people in their car stopping off at the side of the road for a picnic, perhaps changing and discarding a spark plug and leaving some used oil on the ground. The aliens are so far above us, they don’t even think to make contact, and they fail to consider the societal and ecological impact of the items left in the Zone.

Roadside Picnic is a 1971 Russian science fiction novel written by Arkady and Boris Strugatsky and translated to English by Antonina W. Bouis. It tells the story of the effects of the "Visitation," an event during which extraterrestrial visitors spent two days in six different spots around the earth. The title refers to a metaphor used within the story to describe the alien activity in relation to humanity; extraterrestrials briefly paused on Earth for a "roadside picnic," leaving behind their garbage and other materials much as humans might abandon their own clutter along the side of the road. 
There are six "Zones" throughout the Earth, each corresponding to one of the extraterrestrial landing sites. More properly termed Visitation Zones, each Zone is the site of strange artifacts and inexplicable phenomenon, which scientists, governments, and amateur hunters explore and attempt to understand. The importance of these Zones is described in the novel's introductory interview with the character Dr. Pilman:

After all, many very important scientists have proposed that the discoveries made in the Visitation Zones are capable of changing the entire course of our history.

Zones are often explored by groups of people called "stalkers," who dangerously and illegally enter them to essentially hunt for alien artifacts. The stalkers and their activities form the basis for the plot of the novel. 
Within the course of the story, the Zone specifically focused on is in Harmont, Canada. The stalker Red Schuhart makes a series of expeditions into this Zone over an eight-year period. It is full of objects that seem to defy all logic, including hazardous materials, legendary objects that can grant wishes, and others whose purposes are not known. 

a_n = n*e^(-n/2) Determine whether the sequence with the given n'th term is monotonic and whether it is bounded.

a_n=n e^(-n/2)
Monotonicity
First  five terms of the sequence are
a_1=e^(-1/2)=0.6065
a_2=2e^-1=0.7358
a_3=3e^(-3/2)=0.6693
a_4=4e^-2=0.5413
a_5=5e^(-5/2)=0.4104
We can see that after the second term, the terms are decreasing so it is possible that the whole sequence is monotonically decreasing. Let us verify that.
a_n>a_(n+1)
n e^(-n/2)>(n+1)e^(-(n+1)/2)=(n+1)e^(-n/2)e^(-1/2)
Divide the inequality by e^(-n/2). We can do that because e^(-n/2)>0, forall n in NN.   
n>(n+1)e^(-1/2)=n e^(-1/2)+e^(-1/2)
Divide by n. We can do that because n>0.
1>e^(-1/2)+e^(-1/2)/n
Since e^(-1/2)<1, we can find sufficiently large n such that the above inequality holds. In this case n=2.
1>e^(-1/2)+e^(-1/2)/2=0.9098
Therefore, forall n geq 2 (a_n>a_(n+1)) which means that the sequence is monotonically decreasing.
Boundedness
We have shown that the sequence is monotonically decreasing from second term onwards. This means that the second terms is also maximum of the sequence. In other words the sequence is bounded from above by a_2=2e^-1.
On the other hand if we look at the sequence a_n=n e^(-n/2), we see that all of its terms are positive. This is because n>0 and exponential function is always positive so e^(-n/2)>0 and the product of two positive numbers is itself positive.
Therefore, a_n>0, forall n in NN i.e. the sequence is bounded by zero from below.
We can conclude that forall n in NN, a_n in [2e^(-1),0) i.e. the sequence is bounded from both below and above.                                                         
The image below shows first 20 terms of the sequence. Both boundedness and monotonicity can clearly be seen on the image.                    
 

Why should the Senate defer to the president's choice of who he wants working under him?

There are reasons why the Senate should defer to the choices the President makes regarding whom he wants to work with him. It is important for the President to be able to choose people with whom he will be comfortable working. These people will help the President carry out his vision, and they should be people who are generally supportive of the President and his ideas and goals. It is important, for example, that the President and the Secretary of State share the same goals for American foreign policy. The Senate should not be an obstruction to the President; by not allowing the President to choose a Secretary of State with whom he is comfortable, the Senate potentially impedes efficacy of the executive branch.
It is also customary for the Senate to give the President an opportunity to implement some of his ideas and policies. To do this, he will need people who share his vision, ideas, and goals. Finally, when the President is successful in accomplishing goals that are beneficial for the country, the country benefits. The Senate should be supportive in allowing the President an opportunity to accomplish his goals that benefit the country. One way to assist with this is to have people in place with whom the President feels comfortable.

In The Crucible, who called Elizabeth out and why?

According to Cheever, Abigail Williams made the claim that Elizabeth's spirit stuck needles into her (Abigail's) stomach. Cheever is then summoned to the Proctor house to question Elizabeth and look for evidence of this type of witchcraft. He finds a poppet (doll) with needles stuck in it. This is what Abigail claimed. She concluded that Elizabeth stuck needles in the doll and used her (voodoo) powers of witchcraft to inflict that violence on Abigail herself.
Of course, this is untrue. Abigail is pretending, and she had a role in using Mary Warren to plant the doll (poppet) in Elizabeth's home. As a result of Abigail's accusation and finding the poppet in her home, Cheever is forced to arrest Elizabeth.
Abigail accuses Elizabeth out of jealousy. She and John Proctor, Elizabeth's husband, had an affair, but John ended it. Abigail, still in love with John, did not want their relationship to end. When he abandons her, she takes her revenge upon Elizabeth. At this point, John Proctor's only hope is to convince Mary Warren to admit it was her poppet and the accusations of witchcraft are all false. John is outraged that his wife has been charged, but he must know his affair with Abigail indirectly contributed to his wife's arrest.

Calculus of a Single Variable, Chapter 9, 9.6, Section 9.6, Problem 21

When using Root test on a series sum a_n , we determine a limit as:
lim_(n-gtoo) root(n)(|a_n|)= L
or
lim_(n-gtoo) |a_n|^(1/n)= L
Then, we follow the conditions:
a) Llt1 then the series is absolutely convergent.
b) Lgt1 then the series is divergent.
c) L=1 or does not exist then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
We may apply the Root Test to determine the convergence or divergence of the series sum_(n=1)^oo n^3/3^n . when we let: a_n =n^3/3^n .
Applying the Root test, we set-up the limit as:
lim_(n-gtoo) |n^3/3^n|^(1/n) =lim_(n-gtoo) (n^3/3^n)^(1/n)
Apply Law of Exponents: (x/y)^n = x^n/y^n and (x^n)^m= x^(n*m) .
lim_(n-gtoo) (n^3/3^n)^(1/n) =lim_(n-gtoo) (n^3)^(1/n)/(3^n)^(1/n)
=lim_(n-gtoo) n^(3*1/n)/3^(n*1/n)
=lim_(n-gtoo) n^(3/n)/3^(n/n)
=lim_(n-gtoo) n^(3/n)/3^1
=lim_(n-gtoo) n^(3/n)/3
Evaluate the limit.
lim_(n-gtoo)n^(3/n)/ 3 =1/3 lim_(n-gtoo) n^(3/n)
=1/3 *1
=1/3
The limit value L =1/3 satisfies the condition: Llt1 since 1/3lt1 .
Therefore, the series sum_(n=1)^oo n^3/3^n is absolutely convergent.