sum_(n=1)^oo (lnn/n)^n Use the Root Test to determine the convergence or divergence of the series.

To determine the convergence or divergence of a series sum a_n using Root test, we evaluate 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.
For the given series sum_(n=1)^oo (ln(n)/n)^n , we have a_n =(ln(n)/n)^n .
Applying the Root test, we set-up the limit as:
lim_(n-gtoo) |(ln(n)/n)^n|^(1/n) =lim_(n-gtoo) ((ln(n)/n)^n)^(1/n)
Apply Law of Exponent: (x^n)^m = x^(n*m) .
 
lim_(n-gtoo) ((ln(n)/n)^n)^(1/n)=lim_(n-gtoo) (ln(n)/n)^(n*1/n)
                                =lim_(n-gtoo) (ln(n)/n)^(n/n)
                                =lim_(n-gtoo) (ln(n)/n)^1
                               =lim_(n-gtoo) (ln(n)/n)
Evaluate the limit using direct substitution: n = oo .
lim_(n-gtoo) (ln(n)/n) = oo/oo
When the limit value is indeterminate (oo/oo) , we may apply L'Hospital's Rule:
lim_(x-gta) (f(x))/(g(x)) =lim_(x-gta) (f'(x))/(g'(x)) .
Let: f(n) = ln(n) then 
       g(n) = n then g'(n) =1 .
Then, the limit becomes:
lim_(n-gtoo) (ln(n)/n)=lim_(n-gtoo) ((1/n))/1
                      =lim_(n-gtoo) 1/n
                      = 1/oo
                      =0
The limit value   L=0 satisfies the condition: L lt1 since 0lt1.
Therefore, the series  sum_(n=1)^oo (ln(n)/n)^n is absolutely convergent.

(dr)/(ds) = 0.75r Find the general solution of the differential equation

The general solution of a differential equation in a form of  can be evaluated using direct integration. The derivative of y denoted as y' can be written as (dy)/(dx) then y'= f(x) can be expressed as (dy)/(dx)= f(x)
For the problem (dr)/(ds)=0.75r , we may apply variable separable differential equation in which we set it up as f(y) dy= f(x) dx .
Then,(dr)/(ds)=0.75r can be rearrange into (dr)/r=0.75 ds .
 
Applying direct integration on both sides:
int (dr)/r= int 0.75 ds .
For the left side, we apply the basic integration formula for logarithm: int (du)/u = ln|u|+C
int (dr) /r = ln|r|
For the right side, we may apply the basic integration property: int c*f(x)dx= c int f(x) dx .
int 0.75 ds=0.75int ds .
Then the indefinite integral will be:
0.75int ds= 0.75s+C
Combining the results for the general solution of differential equation: 
ln|r|=0.75s+C
 r= Ce^(0.75s)

y = 4 - x^2 Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

To find the volume of a solid by revolving the graph of y =4-x^2 about the x-axis, we consider  the bounded region in between the graph and the x-axis. To evaluate this, we apply Disk method  by using a rectangular strip  perpendicular to the axis of rotation. As shown on the attached image, we consider a vertical rectangular strip with a thickness =dx. 
We follow the formula for  the Disk Method in a form of: V = int_a^b pir^2 dx or V = pi int_a^b r^2 dx
 where r is the length of the rectangular strip.
 In this problem, we let the length of the rectangular strip=y_(above)-y_(below) .
 Then r = (4-x^2) - 0 = 4-x^2
Boundary values of x: a= -2 to b=2 .
Plug-in the values on the formula V = pi int_a^b r^2 dx , we get:
V =pi int_(-2)^2 (4-x^2)^2 dx
Expand using FOIL method:(4-x^2)^2 = (4-x^2)(4-x^2)= 16-8x^2+x^4 .
 The integral becomes:
V =pi int_(-2)^2 (16-8x^2+x^4) dx
Apply basic integration property:int (u+-v+-w)dx = int (u)dx+-int (v)dx+-int(w)dx  to be able to integrate them separately using Power rule for integration:  int x^n dx = x^(n+1)/(n+1) .
V = pi[ int_(-2)^2(16) dx -int_(-2)^2(8x^2)dx+int_(-2)^2(x^4)dx]
V = pi[16x-(8x^3)/3+x^5/5]|_(-2)^2
Apply definite integration formula: int_a^b f(y) dy= F(b)-F(a) .
V = pi[16(2)-(8(2)^3)/3+(2)^5/5]-pi[16(-2)-(8(-2)^3)/3+(-2)^5/5]
V =pi[32-64/3+32/5]-pi[-32-(-64)/3+(-32)/5]
V =pi[32-64/3+32/5]-pi[-32+64/3-32/5]
V=(256pi)/15 -(-256pi)/15
V=(256pi)/15 +(256pi)/15
V=(512pi)/15 or 107.23 (approximated value)
 

I need to compose "The Sirens" that is personalized to my dream, which is to major aviation in college and become an airline pilot in 4 stanzas of at least 4 lines. Please show me the example.

It sounds like the assignment you're describing is to compose a poem about your personal goals using the mythical sirens as a motif. 
The sirens of the Odyssey and Greek myth were creatures often described as part woman part bird. With their alluring song, they shipwrecked sailors on their rocky island. When Odysseus encounters the sirens in Homer's Odyssey, he is curious about their song. He instructs his crew to stop up their own ears with beeswax, so they will be unaffected while he is tied to the mast of the ship. This way, Odysseus will not be able to act upon the siren's call. 
I suggest pre-writing ideas for your poem by listing those "siren songs" that could possibly distract you from your ultimate goal of majoring in aviation in college. A possible organizational strategy could be to make each stanza a distinct siren call that could tempt you away from your dream. You could also imagine what form these sirens would take. For example, a siren call could be lack of focus or your own inner voice that doesn't want to risk failure. 
You should also consider the point of view you would like to write from. You could write from the perspective of the siren(s) or from your own point of view. You could also think about the time frame you'd like to write from. Are you imagining the siren calls that could come in the future, currently trying to block out their song, or thinking back on the siren calls that threatened to throw you off course? 

Who is Mary Carson in The Thorn Birds most like in the Kite Runner and A Thousand Splendid Suns?

Mary Carson in The Thorn Birds is Paddy's sister who offers Paddy a job at Drogheda. She is an imperious, controlling widow who convinced her late husband, Michael Carson, to marry her because she was a good-looking woman with brains in Australia, where there were few women. The author describes Mary as wielding as much power "as any puissant war lord of elder days." She controls people in the local society with her imperious ways.
In some ways, Mary might be compared to Sanaubar in The Kite Runner. Sanaubar is a beautiful but notoriously "unscrupulous" woman who marries Ali, 19 years her senior. Like Ali, she is a Hazara and a Shi'a. She has beautiful green eyes that have apparently tempted many men. Sanaubar, like Mary Carson, is not afraid to speak her mind, and she openly expresses her disdain for Ali, her disfigured husband. Like Mary Carson, Sanaubar is married to a man who is less clever and meeker than she is, and Sanaubar runs off five days after her son, Hassan, is born. She is, like Mary Carson, powerful and unkind.
It could be argued that Laila in A Thousand Splendid Suns does not resemble Mary Carson at first but comes to be more like her over time. Laila, who is beautiful and brilliant like Mary Carson, marries Rasheed because she can benefit in a material sense from the marriage. She is much younger than Rasheed, but she is pregnant and needs a father for her child. As she seeks to gain an advantage from her marriage, she is like Mary Carson. She, like Mary, becomes a widow when Mariam, Rasheed's first wife, kills Rasheed. Laila then has more power and is reunited with her true love, Tariq. Eventually, Laila becomes better able to steer her own destiny as Mary Carson is. You may, however, decide that other characters are more like Mary Carson.

y = 1/2(e^x + e^(-x)) , [0, 2] Find the arc length of the graph of the function over the indicated interval.

The arc length of the curve y=f(x) between x=a and x=b, (aL=int_a^b sqrt(1+y'^2)dx
Before we start using the above formula let us notice that y=1/2(e^x+e^-x)=cosh x. This should simplify our calculations.
L=int_0^2sqrt(1+(cosh x)'^2)dx=
int_0^2sqrt(1+sinh^2x )dx=
Now we use the formula cosh^2 x=1+sinh^2x.
int_0^2cosh x dx=
sinh x|_0^2=
sinh 2-sinh 0=
sinh 2 approx3.62686   
The arc length of the graph of the given function over interval [0,2] is sinh 2 or approximately 3.62686.
The graph of the function can be seen in the image below (part of the graph for which we calculated the length is blue).
https://en.wikipedia.org/wiki/Hyperbolic_function

Single Variable Calculus, Chapter 7, Review Exercises, Section Review Exercises, Problem 40

Differentiate $\displaystyle y = \arctan (\arcsin \sqrt{x})$


$
\begin{equation}
\begin{aligned}

y' =& \frac{d}{dx} [\arctan (\arcsin \sqrt{x})]
\\
\\
y' =& \frac{1}{1 + (\arcsin \sqrt{x})^2} \cdot \frac{d}{dx} (\arcsin \sqrt{x})
\\
\\
y' =& \frac{1}{1 + (\arcsin \sqrt{x})^2} \cdot \frac{1}{\sqrt{1 - (\sqrt{x})^2}} \frac{d}{dx} (\sqrt{x})
\\
\\
y' =&\frac{1}{1 + (\arcsin \sqrt{x})^2} \cdot \frac{1}{\sqrt{1 - x}} \frac{d}{dx} (x^{\frac{1}{2}})
\\
\\
y' =& \frac{1}{\sqrt{1 - x} [1 + (\arcsin \sqrt{x})^2]} \cdot \frac{1}{2} x^{\frac{-1}{2}}
\\
\\
y' =& \frac{1}{2 x^{\frac{1}{2}} \sqrt{1 - x} [1 + (\arcsin \sqrt{x})^2] }
\\
\\
y' =& \frac{1}{2 x^{\frac{1}{2}} (1 - x)^{\frac{1}{2}} [1 + (\arcsin \sqrt{x})^2] }
\\
\\
y' =& \frac{1}{2 (x - x^2)^{\frac{1}{2}} [1 + (\arcsin \sqrt{x})^2]}
\\
\\
& \text{ or }
\\
\\
y' =& \frac{1}{2 \sqrt{x - x^2} [1 + (\arcsin \sqrt{x})^2]}



\end{aligned}
\end{equation}
$