Let f be defined by the function f(x) = 1/(x^2+9)
(a) Evaluate the improper integral [tex]\int\limits^{∞}_3 {f(x)} \, dx[/tex] or show that the integral diverges
(b) Determine whether the series ∑n=3∞ f(n) converges or diverges State the conditions of the test used for determining convergence or divergence
(c) Determine whether the series ∑n=1∞(−1)n(en⋅f(n))=∑n=1∞(−1)n(n2+9)en converges absolutely, converges conditionally, or diverges (image put below)

(a)

[tex]\displaystyle\int_3^\infty \frac{\mathrm dx}{x^2+9}=\lim_{b\to\infty}\int_{x=3}^{x=b}\frac{\mathrm dx}{x^2+9}[/tex]

Substitute x = 3 tan(t ) and dx = 3 sec²(t ) dt :

[tex]\displaystyle\lim_{b\to\infty}\int_{t=\arctan(1)}^{t=\arctan\left(\frac b3\right)}\frac{3\sec^2(t)}{(3\tan(t))^2+9}\,\mathrm dt=\frac13\lim_{b\to\infty}\int_{t=\arctan(1)}^{t=\arctan\left(\frac b3\right)}\mathrm dt[/tex]

[tex]=\displaystyle \frac13 \lim_{b\to\infty}\left(\arctan\left(\frac b3\right)-\arctan(1)\right)=\boxed{\dfrac\pi{12}}[/tex]

(b) The series

[tex]\displaystyle \sum_{n=3}^\infty \frac1{n^2+9}[/tex]

converges by comparison to the convergent p-series,

[tex]\displaystyle\sum_{n=3}^\infty\frac1{n^2}[/tex]

(c) The series

[tex]\displaystyle \sum_{n=1}^\infty \frac{(-1)^n (n^2+9)}{e^n}[/tex]

converges absolutely, since

[tex]\displaystyle \sum_{n=1}^\infty \left|\frac{(-1)^n (n^2+9)}{e^n}\right|=\sum_{n=1}^\infty \frac{n^2+9}{e^n} < \sum_{n=1}^\infty \frac{n^2}{e^n} < \sum_{n=1}^\infty \frac1{e^n}=\frac1{e-1}[/tex]

That is, ∑ (-1)ⁿ (n ² + 9)/eⁿ converges absolutely because ∑ |(-1)ⁿ (n ² + 9)/eⁿ| = ∑ (n ² + 9)/eⁿ in turn converges by comparison to a geometric series.

which one ??????????????????????????

in certain mill the cost of c in thousands of dollars of producing x tons of steel is given by c(x)= 0.03x +3.4. the revenue R in thousands of dollars by selling x tons is given R(x)= 0.5x. Find x the number of tons that must be produced to break even

Find the area of the figure.
A) 96cm
B) 110cm
C) 120cm
D) 150cm​

I need a answer ASAP​

Camden is going to invest $600 and leave it in an account for 12 years.
Assuming the interest is compounded continuously, what interest rate, to the
nearest hundredth of a percent, would be required in order for Camden to
end up with $920?

can someone help me with this please​

PLEASE HELP ITS DUE IN AN HOUR!!!!!!

Calculate the volume of this triangular prism?

NEED HELP PLEASE

We described numbers in this lesson using both powers of 10 and using standard decimal value. For example, the speed of light through ice can be written as a multiple of a power of 10, such as (2.3)⋅108 meters per second, or as a value, such as 230,000,000 meters per second. Use the number line to answer questions about points A and B.
Describe point B as:

a. A multiple of a power of 10
____ x 10 ^ ____
b. A value
_______

Describe point A as:

a. A multiple of a power of 10
_____ x 10 ^ _____

b. A value
______

What is 18.5% of 500?

Axis of sym: x =

Vertex: (
,
)

Do not do the domain

Range: Pick one: A. y≥4 B. y<4 C. y≤4 D. y≤−2 E. y<−2

The Range is

Zeros:
and


Equation:

Each pattern below is made of six congruent
squares. Which pattern can be folded into a cube
that has a surface area of 96cm??

simplify the expression by using the distributive property 3(7y+4)​

Wayne charges the following for repairing washing machines:
£28 call-out charge + £16 for each half-hour he spends on the repair
If a repair costs £76, how long did it take?

Eloise spent $1.65 on potato salad that costs $0.25 per pound.

How many pounds of potato salad did she buy?

A hipopótamos weight 1.5 tons what is its weight in ounces

what name of triangle name is when they only have two side the same lenght

Let X and Y be the following sets:
X = {}
Y = {2,3, 5, 7, 11, 13}
Which of the following is the set X Y?

For keeping business rocords, every three mouths of a year is called a quater. How many months are equal to three-quaters of a year? Explain how you found your answer.​

what is the value of angle x?​

mr kabi invest k375 at a simple interest rate of 8.375 per annum what amount will his investment grow to over 4 years 8 months?​

factor g(x) =6x^3 +13x^2 -2x -1 if (2x + 3) is a factor.​

Jacque was selling bottles of water at the golf tournament Between 7 am and 8 am he sold 2 bottles of water, between 8 am and 9 am he sold 8 bottles, and between 9 am and 10 am he sold 32 bottles. If this trend
continues, how many bottles of water will he sell between 10 am and 11 am?
32
64
128
256

El precio de cada acción de una empresa de tecnología ha presentado a través del tiempo un comportamiento dado por la siguiente función:



Donde t es el tiempo dado en días y A es el precio de la acción inicial cuando t = 0. Un inversionista desea conocer el número de días para el cual el valor de la acción se duplica. Para conocer este resultado, el proceso que se debe realizar es reemplazar f(t) por


A.
A/2, luego, expresar la ecuación en forma de logaritmo natural y ahora despejar t.

B.
2A, luego, expresar la ecuación en forma de logaritmo natural y ahora despejar t.

C.
2A, luego, dividir ambos lados de la ecuación por e y ahora despejar t.

D.
A/2, luego, dividir ambos lados de la ecuación por e y ahora despejar t.

Find the unit rate.. If a family travels in their mini-van for 465 miles and uses 15 gallons of gas what is the mileage per gallon?
10 points

jeremiah has a total of 144 pokemon cards 40 out of every 48 cards are not holographic cards at this rate how many of jeremiah cards are holographic

(x+6)/(x+1)=6/2
Solving proportions

What is the volume of the figure shown below?

help!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Please help!! No links