Advanced Placement (AP) Middle School

100 Points [Calculus] Finding Power Series. An explanation would be greatly appreciated. See attached image.

Answers

Answer 1

It’s the third option. Also look at the explanation

Answer 2

Answer:

[tex]\textsf{C)}\quad \displaystyle 2\sum^{\infty}_{n=0} \dfrac{x^{2n+1}}{(2n+1)!}[/tex]

Explanation:

The series expansions for [tex]e^x[/tex] and [tex]e^{-x}[/tex] are:

[tex]\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}=1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+ \cdots \\\\\\\\e^{-x} = \sum_{n=0}^{\infty} \frac{(-x)^n}{n!}=1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}- \cdots[/tex]

Therefore:

[tex]\displaystyle e^x -e^{-x} = \left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\right)-\left(1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!} -\cdots\right)[/tex]

This simplifies to:

[tex]\displaystyle e^x -e^{-x} = \left(x + \frac{x^3}{3!} + \frac{x^5}{5!}+\cdots\right)-\left( - x - \frac{x^3}{3!} -\frac{x^5}{5!} -\cdots\right)[/tex]

[tex]\displaystyle e^x -e^{-x} = \left(x + \frac{x^3}{3!} + \frac{x^5}{5!}+\cdots\right)+\left( x + \frac{x^3}{3!} +\frac{x^5}{5!} +\cdots\right)[/tex]

[tex]\displaystyle e^x -e^{-x} =2 \left(\dfrac{x^1}{1!} + \frac{x^3}{3!} + \frac{x^5}{5!}+\cdots\right)[/tex]

The expression inside the parentheses represents terms where the numerators are x raised to consecutive positive odd numbers, and the denominators are factorials of those odd numbers

A positive odd number can be expressed as 2n + 1, where n is a non-negative integer starting from zero.

So, the numerator of each term in the series can be expressed as [tex]x^{2n+1}[/tex] and the denominator as (2n + 1)!.

Therefore, we can express the power series of [tex]e^x -e^{-x}[/tex] as:

[tex]\displaystyle e^x -e^{-x}=2\sum^{\infty}_{n=0} \dfrac{x^{2n+1}}{(2n+1)!}[/tex]

Related Questions

Tom scored 200 on a test with a mean of 150 and a standard deviation of 25. What is Toms z score and percentile?

Answers

Answer:

Z score is 2.0

In normal distribution Percentile is

97. 72% Left tailed

2.28% Right tailed

Explanation:

Z score is = (score - mean) / SD

Z = (200-150)/25=50/25=2

The value of Tom's z-score and percentile are;

Z-score = 2

Percentile = 97.725th percentile.

We are given;

score; x' = 200

Mean score; μ = 150

Standard deviation; σ = 25

Formula for z-score here is;

z = (x' - μ)/σ

z = (200 - 150)/25

z = 2

From p-value from z-score calculator, the percentile of a z-score of 2 is 0.97725 or 97.725th percentile.

Read more about z-score at; https://brainly.com/question/25638875

When you can tell how a person feels based on the person's facial expression, which of the four components of emotional intelligence

propagated by Mayer and Salovey comes into play?

A understanding emotions

B. perceiving emotions

C managing emotions

D. using emotions

Answers

Answer:

perceiving emotions

Explanation: got it right on my test!

On the basis of the person's facial expression, you can tell how a person feels by perceiving their emotions. Thus, the correct option for this question is B.

What are the four components of emotional intelligence?

The four components of emotional intelligence are as follows;

Self-awareness.Self-management.Social awareness.Relational management.

Apart from this emotional intelligence, some other components are also essential in determining the feeling of an individual. They may include empathy, motivation, social skills, etc.

Among the given four components of emotional intelligence, you can definitely tell the feeling of an individual by analyzing their facial expression through the perception of their thoughts, emotions, ideas, etc.

Therefore, you can tell how a person feels by perceiving their emotions. Thus, the correct option for this question is B.

To learn more about Emotional intelligence, refer to the link:

https://brainly.com/question/7905042

#SPJ5

Inorganic mercury is not particularly harmful, but its release into the environment can be hazardous because of a chemical transformation it undergoes into the highly toxic substance known as _______________________________________

Answers

Answer : The kind of substance is probably “Methylmercury.”

Which of the following is air that moves from regions of high pressure to regions of low pressure?
Orain
wind
snow
clouds

Answers

Clouds, correct me if I’m wrong, sorry

Every student in fourth grade at Watson Elementary took a Big Five Personality test the first week of school. Principal Albert
recorded the agreeableness score for each child along with the number of referrals for misconduct. He thinks that rating high on
the agreeableness scale might predict better behavior from students.
What type of graph should he choose to best represent his data?

Answers

Answer: Scatterplot

Explanation: The scatterplot is the appropriate way to graph a correlational set of data to display the direction and strength of a relationship.

Answer:

scatter plot

Explanation:

How do folk and popular cultures differ in the ways they help form a society’s overall culture?

Answers

Answer:

Folk culture and popular culture differ in their patterns of origins,distribution and diffusion. Folk culture is practiced by small homogeneous groups living in isolated rural areas while Popular culture is found in large heterogeneous societies that share certain customs despite differences in other personal characteristics among them.

Explanation:

In terms of the way they are both different from each other, we can explain under the patterns listed i.e origin, distribution and diffusion.

Origin: Customs originate from hearths. Folk customs are often anonymous while popular customs originate in more developed countries as part of the market for recreational and disposable income to purchase these material goods.

Distribution :Popular culture is distributed widely across many countries, with little regard for physical factors while Folk cultures often (though not always) incorporate elements of the local environment of wherever it is being noticed. Groups with relatively little contact with others develop unique folk cultures. Thus, people living in some remote environments tend to have a specific culture related to their environment and quite different from other geographical locations practicing same named culture.

Diffusion: Popular culture diffuses or mixes (usually hierarchically) through rapid electronic communications and transportation networks while Folk culture diffuses or mixes through relocation diffusion.

What were nazis taught about other races? How do you think this affected the way that they interacted with people that were different form them?

Answers

Nazis were taught that if someone didn’t look a certain way they weren’t aloud to have what they had, children, respect, good food. And since they were taught this way it affected the people around them by scaring them because they thought they were better than everyone else.

Which term best describes the type of work done by a postindustrial
community?
O
A. Administration
O
B. Hunting and gathering
O
C. Agriculture
O
D. Industry

Answers

Answer:

Explanation:

because postindustrial don't do any of the other's

Humans often convert grosslands into farm fields to grow crops. Describe why the conversion of temperate grasslands, such as in the central United States, into agricultural areas is an
environmental problem.

Answers

Answer and Explanation:

Grasslands are highly important habitats for hundreds to thousands of ecologically important species of fauna and flora that are adapted to the extreme conditions that occur here. However, they are currently threatened by human activities such as habitat destruction and fragmentation for their transformation into agricultural areas.

This disturbance can lead to the alteration of the soil if agricultural practices are not carried out properly, which would have detrimental consequences for the growth of species of flora. Moreover, farmers tend to use a high level of pesticides that, after being carried away by rain or wind, are dangerous for wildlife and the presence of grazing animals can also lead to the destruction of different species of grass that compose the habitat of these organisms. In addition, researchers suggest that, as rainfall patterns change with climate change, these ecosystems could become deserts due to the lack of vegetation.

Grasslands are home to several species hence preservation of grassland maintains biodiversity.

Grasslands are the habitat of so many species such as wildlife, insects, rare plant species, etc. All these organisms constitute the biodiversity that exists in the biosphere.

When this rich ecosystem with such great biodiversity is converted to agricultural farm fields to grow crops, many species may become extinct or forced to relocate due to loss of habitat. This is not good for environmental conservation efforts.

Recently, grasslands have come under pressure owing to population explosion and the need to have more arable land for food production. Several grasslands in the central United States are being convert to grassland with consequent deleterious impact on the environment.

Learn more: https://brainly.com/question/24556237

Which statement about rice production in China and Southeast Asia is correct? A large portion of the rice harvest is used to feed animals. Nearly the entire rice crop is consumed domestically. The majority of rice produced is grown by subsistence farmers. High subsidies are needed to make rice production a sustainable venture. Large surpluses of rice are exported to Africa, Europe, and the Middle East.

Answers

Answer:

Large supply of rice is exported to Africa, Europe and the Middle East. I think.

Which two words are synonyms of the word protest? strike snarl march pry jump

Answers

Answer:

Strike and March

Answer:

The answer is STRIKE AND MARCH because them are the two words that are a synonyms HOPE this Helps!!

Explanation:

Describe an example of a technology or team and technology interaction that you have had in the context

of school or work

Answers

The 'middle-class squeeze' in relation to student loan debt is attributed to flat incomes, rising education costs, higher unemployment rates since the 2008 recession, and a lack of savings compared to past generations.

Explanation:Reasons Behind the 'Middle-Class Squeeze' in Relation to Student Loan Debt

There are several reasons behind what researchers refer to as the 'middle-class squeeze' when it comes to student loan debt. Notably, it is caused by a combination of flat incomes and skyrocketing education costs, which lead to a significant financial burden on students and graduates. This situation is exacerbated by challenging economic factors, such as the increase in unemployment since the 2008 recession, creating fewer job opportunities for recent graduates. This makes the repayment of student loans even more daunting, particularly when starting on entry-level wages.

Moreover, the rising tuition rates force students to take on larger loans, resulting in average debts of $25,250 upon graduation as reported in 2010. The interplay between stagnant wages and increasing costs of living compounds the issue, as American families have less money saved compared to previous generations. The phenomenon of the 'middle-class squeeze' is further supported by indicators such as lower bond returns and a widening gap between lower and middle-class incomes. In effect, the social challenge of student loan debt not only impacts the individual graduates but also affects the broader society.

The middle-class squeeze in relation to student loan debt is due to rising tuition rates, stagnating wages, higher private loan interest rates, and limited governmental support for higher education costs, resulting in financial hardships for graduates.

The term "middle-class squeeze" in relation to student loan debt refers to the financial hardship faced by middle-income earners due to the disparity between stagnant wages and the escalating costs of higher education. This has led to a significant burden of debt on graduates. The reasons behind this include rising tuition rates, the increase of student loan debt, limited government funding, higher interest rates on private loans, and sluggish wage growth.

Research by economist Susan Dynarski points out that while the government provides student loans, the structure of the US loan market involves both public and private sectors. Private loans often carry fewer consumer protections and come with higher interest rates, contributing to financial strain. Additionally, Daniela Senderowicz emphasizes that flat incomes and inflating education expenses lead many students towards financial ruin, though she also highlights activism and community building as potential solutions to manage crippling student debts.

Data from the Levy Economics Institute and various studies, like the Class of 2017 college debt statistics, corroborate the issue, revealing the ongoing trend of indebtedness among graduates which impacts the broader economic participation of the middle class.

What is the cost to attend Merced College?

Answers

Answer:

In state tuition 1150 USD, out of state tuition 6115 USD

Explanation:

Answer:

Explanation:

1150

Explanation:

It feels as though some of the information is missing. So, based upon the information provided above, this is the best equation I can come up with. The total amount is up to $20 (but not including $20, so it is an inequality, meaning that $20 cannot be used). 20 is greater than the sum of the number of bottles of w (water) multiplied by the $2 (cost of water) and the number of bottles of j (orange juice) multipled by the $4 (cost of the orange juice).

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