Starting from the geometric series, use power series operations to
determine the Maclaurin series. See picture
Answers
a. By replacing [tex]x[/tex] with [tex]-x^2[/tex] in the power series, we get
[tex]\displaystyle \frac1{1+x^2} = \sum_{n=0}^\infty (-x^2)^n = \sum_{n=0}^\infty (-1)^n x^{2n}[/tex]
Integrate both sides to recover [tex]\arctan(x)[/tex] on the left.
[tex]\displaystyle \int \frac{dx}{1+x^2} = \int \sum_{n=0}^\infty (-x^2)^n = \sum_{n=0}^\infty (-1)^n x^{2n} \, dx[/tex]
[tex]\displaystyle \arctan(x) = C + \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n+1} \, dx[/tex]
By letting [tex]x=0[/tex] on both sides, we find [tex]C=0[/tex], so that
[tex]\displaystyle \arctan(x) = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n+1} \, dx[/tex]
Then dividing both sides by [tex]x[/tex] gives
[tex]\displaystyle \boxed{\frac{\arctan(x)}x = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n} \, dx}[/tex]
b. Let [tex]x=\frac1{\sqrt3}[/tex]. Then
[tex]\displaystyle \frac{\arctan\left(\frac1{\sqrt3}\right)}{\frac1{\sqrt3}} = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} \left(\dfrac1\sqrt3\right)^{2n}[/tex]
[tex]\displaystyle \sqrt3 \arctan\left(\frac1{\sqrt3}\right) = \sum_{n=0}^\infty \frac1{2n+1} \left(-\frac13\right)^n[/tex]
Taking the first few terms from the infinite series, we can approximate
[tex]n=0 \implies \sqrt3\arctan\left(\dfrac1{\sqrt3}\right) \approx 1[/tex]
[tex]0\le n\le1 \implies \sqrt3\arctan\left(\dfrac1{\sqrt3}\right) \approx 1+\dfrac13\left(-\dfrac13\right) = \dfrac89[/tex]
which together suggest the value we want is bounded between 8/9 and 9/9 = 1, hence [tex]\boxed{p=8}[/tex].
Since the series is alternating and converges on [tex]-1<x<0\cup0<x<1[/tex],
[tex]\displaystyle \left| \sum_{n=0}^\infty a_n - \sum_{n=0}^0 a_n \right| < |a_1| = \left|\frac13 \left(-\frac13\right)^1\right| = \frac19[/tex]
and
[tex]\displaystyle \left| \sum_{n=0}^\infty a_n - \sum_{n=0}^1 a_n \right| < |a_2| = \left|\frac15 \left(-\frac13\right)^2\right| = \frac1{45}[/tex]
which tells us the first approximation is off by at most 1/9 from the actual value of [tex]\frac\pi{2\sqrt3}[/tex], whereas the second approximation is off by at most 1/45 from the actual value. In other words, the second approximation is closer, so [tex]\frac\pi{2\sqrt3}[/tex] is closer to [tex]\frac p9[/tex] :
[tex]\dfrac89 \approx 0.8889[/tex]
[tex]\dfrac{\pi}{2\sqrt3}\approx0.9069[/tex]
[tex]\dfrac99 = 1[/tex]
Related Questions
On Monday, a local hamburger shop sold a combined total of 336 hamburgers and cheeseburgers. The number of cheeseburgers sold was three times the number of hamburgers sold. How many hamburgers were sold on Monday
Answers
Answer:
84
Step-by-step explanation:
Let the number of hamburgers be h and the number of cheeseburgers be c.
This means that:
h+c=336c=3hSubstituting c=3h into the first equation, it follows that 4h=336, and thus h=84.
Question 24: [1 + 1 + 1 + 1= 4 points]
Let X be a discrete random variable with the following
x 0.2 0.4 0.5 0.8 1
P(x) 0.1 0.2 0.2 0.3 0.2 (a) Find () , the range of the random variable X
Answers
Considering the given discrete probability distribution, the range of the variable X is given by:
Range(X) = {0.2, 0.4, 0.5, 0.8, 1}.
What is the range of a random variable?The range of a random variable is the set of all values that the variable can assume.
From the table, we have that X can assume the values of 0.2, 0.4, 0.5, 0.8 and 1, hence the range of the variable X is given by:
Range(X) = {0.2, 0.4, 0.5, 0.8, 1}.
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Three books are what percent of four books?
Answers
If four books are 100%, then 3 books are how much percent of total books?
[tex]\frac{3}{4}[/tex] × 100 = 3 × 25 = 75%
Hope it helps!
The answer is 75%.
To find the percentage, divide the number of given books by total books and multiply by 100%.
(3 ÷ 4) × 100%0.75 × 100%75%how many decimal places does 22.22105 have
Answers
Answer:
5
Step-by-step explanation:
where there is the decimal point,you count the numbers after the decimal point
I have 3 more then twice as many socks then keith she has 19 pairs of socks how many do keith have using algebra to solve this what is one step to take?
Answers
Answer:
40
Step-by-step explanation:
19*2 = 38 + 2 = 40
Answer:
Step-by-step explanation:
we have to get k by itself on one side of the equal sign. To do that we have to get rid of 3. To get rid of 3 we have to do the opposite. The opposite of +3 is -3 so we -3 on both sides. 19 - 3 = 16. So k = 16
k+3=19
-3 -3
k=16
Multiply each of the following(X+7)and(y-7)
Answers
Answer:
xy-49
Step-by-step explanation:
x x y is xy and -7x7 is 49 so the answer is xy- 49
A book sold 42,600 copies in its first month of release. Suppose this represents 9.2% of the number of copies sold to date. How many copies have been sold to date?
Answers
The number of copies sold (x) to date is 463043
How to determine how many copies have been sold to date?The given parameters are:
Copies sold = 42,600
Proportion sold = 9.2% of the number of copies sold to date.
The number of copies sold (x) to date is calculated as:
Copies sold = Proportion * number of copies sold to date
Substitute the known values in the above equation
9.2% * x = 42,600
Divide both sides by 9.2%
x = 463043
Hence, the number of copies sold (x) to date is 463043
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2x² + 5x +2can you please give me the answer of 2x + 5 x + 2
Answers
Answer:
you better give me brainliest
Step-by-step explanation:
2x +5x+2 = 7x+2
66 Is 52 more than 7 times a number
Answers
Answer: 2
Step-by-step explanation: 66-52 is 14 and 14/7=2
What is the perimeter of the figure shown? Round your answer to the nearest tenth.
help asap please
tysm
Answers
If the points of the vertices are (-3,1_,(4,2),(9,-3),(2,-4) then the perimeter of the figure is 28.28 units.
Given points of the vertices of the figure having four sides be (-3,1),(4,2),(9,-3),(2,-4).
We have to find the perimeter of the figure.
Perimeter is the sum of lengths of all the sides of a figure.
First give some name to the points so that it will be easy to understand the sides like A(-3,1),B(4,2),C(9,-3),D(2,-4).
Perimeter of the figure =AB+BC+CD+DA
AB=[tex]\sqrt{(2-1)^{2} +(4+3)^{2} }[/tex]
=[tex]\sqrt{1+49}[/tex]
=[tex]\sqrt{50}[/tex] units
BC=[tex]\sqrt{(9-4)^{2} +(-3-2)^{2} }[/tex]
=[tex]\sqrt{25+25}[/tex]
=[tex]\sqrt{50}[/tex]units
CD=[tex]\sqrt{(2-9)^{2} +(-4+3)^{2} }[/tex]
=[tex]\sqrt{49+1}[/tex]
=[tex]\sqrt{50}[/tex] units
DA=[tex]\sqrt{(2+3)^{2} +(-4-1)^{2} }[/tex]
=[tex]\sqrt{25+25}[/tex]
=[tex]\sqrt{50}[/tex] units
Perimeter=[tex]\sqrt{50} +\sqrt{50} +\sqrt{50} +\sqrt{50}[/tex]
=4[tex]\sqrt{50}[/tex]
=4*5*[tex]\sqrt{2}[/tex]
=20[tex]\sqrt{2}[/tex]
=20*1.411
=28.284
After rounding off we will get 28.28 units.
Hence if the points of the vertices are (-3,1_,(4,2),(9,-3),(2,-4) then the perimeter of the figure is 28.28 units.
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If the hypotenuse is c=a√2, then 6=a√2. In inches, what is the value of a?
Answers
The value of a is 3√2 inches if it is given that the hypotenuse is c = a√2 and 6 = a√2. This can be obtained by using Pythagoras' theorem.
Calculate the value of a:This question can be solved using Pythagoras' theorem,
If in a right angled triangle, hypotenuse is the longest side of the triangle and always opposite to the angle 90°, height and base are the two shorter sides which are adjacent to the angle 90°.
If hypotenuse = c, height = a and base = b then,
⇒ hypotenuse² = height² + base²
⇒ c² = a² + b²
Here in the question it is given that,
Hypotenuse of the triangle is c = a√2 6 = a√2 in inches ⇒ c = 6 Height and base is equal and both has the value of a ⇒ height = a and base = aBy using the Pythagoras' theorem we can write that,
⇒ hypotenuse² = height² + base²
⇒ 6² = a² + a²
⇒ 36 = 2a²
By dividing both sides of the equation by 2,
⇒ 18 = a²
⇒ a² = 18
By taking square on both side of the equation,
⇒ a = 3√2 inches
Hence the value of a is 3√2 inches if it is given that the hypotenuse is c = a√2 and 6 = a√2.
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prove 1-cot23 = 2/ ( 1- cot22)
Answers
From all the steps below, we have been able to prove that; 1 - cot23° = 2/(1 - cot22°).
How to prove trigonometric functions?We want to prove that 1 - cot23° = 2/(1 - cot22°).
We will prove it using the trigonometric expression
cot(22° + 23°) = cot45°
Using trigonometric identities, we can rewrite as;
(cot22° cot23° - 1)/(cot22° + cot23°) = 1
Cross multiply to get;
cot22° cot23° - 1 = cot22° + cot23°
Rearrange to get;
cot22° cot23° - 1 - cot22° - cot23° =0
⇒ cot22° cot23° - 1 - cot22° - cot23° + 2 =2
⇒ cot22° cot23° + 1 - cot22° - cot23° =2
⇒ cot22° cot23° - cot22° - cot23° + 1 = 2
⇒ cot22° (cot23° - 1) - 1 (cot23° - 1) = 2
⇒ (cot22° - 1) (cot23° - 1) = 2
Divide both sides by (cot23° - 1) to get;
cot23° - 1 = 2/(cot22° - 1)
⇒ 1 - cot23° = 2/(1 - cot22°).
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Suppose we want to choose 7 colors, without replacement, from 12 distinct colors?
Answers
792 different sets of 7 colors can be chosen.
In how many ways can be the 7 colors selected?Remember that if we have a set of N elements, the number of different sets of K elements that we can make out of these N elements is given by:
C(N, K) = N!/(N - K)!*K!
In this case, we have 12 distinct colors and we want to select 7, so we have:
N = 12
K = 7
Replacing that we get:
C(12, 7) = 12!/(12 - 7)!*7! = 12*11*10*9*8/(5*4*3*2*1) = 792
792 different sets of 7 colors can be chosen.
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2d(3*6)=48y when y=1/4
Answers
Answer: d = 1/3
Step-by-step explanation: 48y = 48/4 = 12. So, 2d(18) is equal to 12. 12/18 is 2/3 so 2d has to equal 2/3. That means d = 1/3.
PLEASE HELP ASAP PLEASE PLEASE
Answers
Answer:
transitive property
Step-by-step explanation:
This is the answer by definition.
The substitution property is demonstrated in the case of m∠AZF=m∠AZL+m∠LZF and m∠AZL+m∠LZF=m∠SZW+m∠LZF, so, m∠AZF=m∠SZW+m∠LZF.
In order to find the value of the unknown, the substitution property is a concept in algebra that is used to substitute the value of a given variable or quantity into an expression.
According to the substitution property of equality, "If two variables x and y are equal, then x can be substituted for y in any equation or expression, and y can be substituted for x in any equation or expression."
According to the question, m∠AZF=m∠AZL+m∠LZF and m∠AZL+m∠LZF=m∠SZW+m∠LZF, so by the substitution property, on substituting the value of m∠AZL+m∠LZF in the first expression, we get, m∠AZF=m∠SZW+m∠LZF.
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Identify the discrete data.
A. the length of a caterpillar during a week
B. the number of dogs you take for a walk each day
C. the wind speed during a particular day
D. the temperature of the water in the local river during the day
Answers
Discrete data exists as a numerical kind of data that contains total, concrete numbers with clear and fixed data values specified by counting.
Therefore, the correct answer is option B. the number of dogs you take for a walk each day.
What is discrete data?Discrete data exists as a numerical kind of data that contains total, concrete numbers with clear and fixed data values specified by counting. Discrete data exists a count that involves integers only a limited number of values exists possible. This kind of data cannot be subdivided into distinct parts. Discrete data contains discrete variables that exist in finite, numeric, countable, and non-negative integers.
Therefore, the correct answer is option B. the number of dogs you take for a walk each day.
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What is the quotient of 7.536\times 10^77.536×10
7
and 7.85 \times 10^27.85×10
2
expressed in scientific notation?
Answers
The value of Quotient from the given terms is 9.6* 10^8
According to the statement
we have given that the two terms and we have to find the quotient of the these terms.
So, we know that the
Quotient is a the number resulting from the division of one number by another.
And for find the quotient we have to divide the terms with each other.
So, The given terms are [tex]7.536*10^7[/tex]and [tex]7.85*10^2[/tex]
divide both terms with each other.
Quotient = [tex]\frac{7.536*10^7}{7.85*10^2}[/tex]
Then
The value of quotient becomes
Quotient = [tex]0.96 * 10^9[/tex]
After removing the decimal from the answer it will become
Quotient = [tex]9.6* 10^8[/tex].
So, The value of Quotient from the given terms is 9.6* 10^8
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Answer: its 7.85x10^-9
Step-by-step explanation:
How many numbers of possible live card hands (hands in five-card poker) drawn without replacement from a standard deck of 52 playing cards?Please give full details!
Answers
The combination shows that the numbers of possible live card hands drawn without replacement from a standard deck of 52 playing cards is 2,598,960.
How to explain the information?A permutation is the act of arranging the objects or numbers in order while combinations are the way of selecting the objects from a group of objects or collection such that the order of the objects does not matter.
Since the order does not matter, it means that each hand is a combination of five cards from a total of 52.
We use the formula for combinations and see that there are a total number of C( 52, 5 ) = 2,598,960 possible hands.
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3,480
A)348 hundreds
B)3,480 ones
C)3 x 1,000+ 4 x 100 + 8 x 80
D)3 thousands and 80 tens
C)3 thousands and 48 tens
Answers
Answer
B
explanation
3480ones = 3480x1=3480
The price of a train ticket consists of an initial fee plus a constant fee per stop. The table compares the number of stops and the price of a ticket (in dollars). Stops Price (dollars) 333 6.506.506, point, 50 777 12.5012.5012, point, 50 111111 18.5018.5018, point, 50 What is the fee per stop?
Answers
Based on the task content given; the initial fee and the fee per stop is $2 and $1.5 respectively.
Equationlet
Initial fee = xFee per stop = yx + 3y = 6.50
x + 7y = 12.50
Subtract both equation to eliminate x7y - 3y = 12.50 - 6.50
4y = 6
y = 6/4
y = 1.5
Substitute y = 1.5 intox + 3y = 6.50
x + 3(1.5) = 6.50
x + 4.5 = 6.50
x = 6.50 - 4.5
x = 2
Therefore, the initial fee and the fee per stop is $2 and $1.5 respectively.
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The letters of "EAGLES" should be evenly spaced across a 70-inch-wide banner, with no margins. Each letter is 10 inches wide. How many inches (x) should exist between each pair of letters
Answers
Answer:
2 Inches or ~1.43 Inches
Step-by-step explanation:
2 Inches if it's not including the spacing before the first "E" and after the "S", ~1.43 Inches if it's including the spacing.
You would get 2 Inches since if each letter is 10 inches (total 6 letters), that would be 60 inches. There are 5 total spaces in between all the letters. 70 Inches(Banner size) - 60 Inches (total letter size) = 10 Inches, divided equally within the 5 spaces. That would equal 2 :)
You also get ~1.43 Inches by doing the same process as the one shown above^ (that is if you're including the spacing before and after the word)
:)
A company that produces dog food claims that its bags contain 11 kg of food. From
prior experimentation, it is known that the weight of the dog food in these bags is normally
distributed with a standard deviation of 0.6 kg. From a random sample of 25 bags, the mean
weight of the dog food was 10.6 kg. We are concerned that the company is exaggerating how
much dog food they put in their bags. Test at the 5% significance level.
Answers
By testing the hypothesis we can conclude that the bag does not contain 11 kg of food.
Given mean of 10.6 kg, population standard deviation of 0.6 and sample size of 25.
We are required to find whether the company is right in saying that their bags contain 11 kg of food.
First we have to make hypothesis.
[tex]H_{0}[/tex]:μ≠11
[tex]H_{1}[/tex]:μ=11
We have to use t statistic because the sample size is less than 30.
t=(X-μ)/s/[tex]\sqrt{n}[/tex]
We will use s/[tex]\sqrt{n}[/tex]=0.6 because we have already given population standard deviation of weights.
t=(11-10.6)/0.6
=0.4/0.6
=0.667
Degree of freedom=n-1
=25-1
=24
T critical at 0.05 with degree of freedom 24=2.0639
T critical at 0.05 with degree of freedom is greater than calculated t so we will accept the null hypothesis.It concludes that the bags donot contain 11 kg of food.
Hence by testing the hypothesis we can conclude that the bag does not contain 11 kg of food.
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Consider the line 3x - 4 y = 7.
What is the slope of a line perpendicular to this line?
What is the slope of a line parallel to this line?
Answers
Answer:
The slope of a line perpendicular to this line is -1/3. The slope of a line parallel to this line would be 3.
Step-by-step explanation:
Perpendicular lines have opposite(sign) and reciprocal slopes.
Parallel lines have the same slope.
I need help please yall
use orders of operation and simplify
Answers
Answer: [tex]\large\boxed{12x^3-30x+3=0}[/tex]
Step-by-step explanation:
[tex]Solve:\\\ 6(2x^2-5)x=-3\\\\Use\ distributive\ property\ first\\\\(12x^2-30)x=-3\\\\Now\ multiply\ the\ terms\ in\ ()\ by\ 'x'\\\\12x^3-30x=-3\\\\\large\boxed{12x^3-30x+3=0}[/tex]
a chord subtend an angle of 72⁰ at the center of a circle with radius 24.5m. calculate the perimeter of the minor segment
Answers
The perimeter of the minor segment is 77. 4m
How to determine the perimeter
The formula for determining the perimeter of the minor segment is given as;
Perimeter = θ/360 × 2πr + 2r sin θ/2
where;
radius = 24. 5mθ = 72Substitute into the formula;
Perimeter = (72/360 × 2 × 3. 142 × 24. 5) +( 2 × 24. 5 × sin 72)
Perimeter = 30. 79 + 46. 60
Perimeter = 77. 4 m
Thus, the perimeter of the minor segment is 77. 4m
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Some of the steps in the derivation of the quadratic formula are shown. Step 4: StartFraction negative 4 a c + b squared Over 4 a EndFraction = a ( x + StartFraction b Over 2 a EndFraction) squared Step 5: (StartFractio.n 1 Over a EndFraction) StartFraction b squared minus 4 a c Over 4 a EndFraction = (StartFraction 1 Over a EndFraction) a (x + StartFraction b Over 2 a EndFraction) squared Step 6: StartFraction b squared minus 4 a c Over 4 a squared EndFraction = ( x + StartFraction b Over 2 a EndFraction) squared Step 7: StartFraction plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFraction = x + StartFraction b Over 2 a EndFraction Which best explains why the expression plus or minus StartRoot b squared minus 4 a c EndRoot cannot be rewritten as b plus or minus StartRoot negative 4 a c EndRoot during the next step? Negative values, like −4ac, do not have a square root. The ± symbol prevents the square root from being evaluated. The square root of terms separated by addition and subtraction cannot be calculated individually. The entire term b2 − 4ac must be divided by 2a before its square root can be determined.
Answers
The reason why we used plus or minus when taking the square root is because; The square root of terms separated by addition and subtraction cannot be calculated individually.
How to use the quadratic formula?
The general formula for a quadratic equation is;
ax² + bx + c = 0
Now, the steps to derive quadratic formula to find the roots of this equation are as follows;
Step 1; Divide both sides of the equation by a to get;
x² + (b/a)x + c/a = 0
Step 2; Transpose the quantity c/a to the right side of the equation to get; x² + (b/a)x = -c/a
Step 3; Complete the square by adding b²/4a² to both sides of the equation to get; x² + (b/a)x + b²/4a² = -c/a + b²/4a²
Step 4; Factor the left side and combine the right side to get;
(x + (b/2a))² = (b² - 4ac)/(4a²)
Step 5: Extract the square-root of both sides of the equation to get;
x + (b/2a) = ±(√(b² - 4ac))/2a
The reason why we used plus or minus when taking the square root is because The square root of terms separated by addition and subtraction cannot be calculated individually.
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In questions 6 – 9, state the solutions for the quadratic equation depicted in the graph.
Answers
Step-by-step explanation:
It is just where it crosses the x axis
6) -3, -4
7) 1, -6
8) -5, -6
9) 3, -2.5
Question 2 Multiple Choice Worth 1 points)
(08.05 MC)
Functions f(x) and g(x) are shown:
f(x)=x²
g(x)=x²-12x+36
In which direction and by how many units should f(x) be shifted to obtain g(x)?
O Left by 18 units
O Right by 18 units
O Left by 6 units
O Right by 6 units
2
Answers
Comparing g(x) with f(x), you can see that the function f(x) is translated to the right by 6 units to produce g(x) which is equivalent to (x-6)²
Transformation of functionTransformation is a techniques use to change the position of an object on an xy-plane.
Given the parent function f(x) = x² and the function g(x) = x²-12x +36
Factorize g(x);
g(x) = x²-6x-6x+36
g(x)=x(x-6)-6(x-6)
Group the terms to have;
g(x) = (x-6)²
Comparing g(x) with f(x), you can see that the function f(x) is translated to the right by 6 units to produce g(x) which is equivalent to (x-6)²
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If the diameter of a circular pond is x meter, what is the its circumference?
Answers
Answer:
piex
Step-by-step explanation:
the circumference of a circle is pie times the diameter of the circle.
the number has 3 digits the number is less than 140 the number has 7 as a factor the number is even the sum of the digits is less than 9
Answers
Answer:
It could be 105, 112 or 133.
Step-by-step explanation:
Having 7 as a factor means it is a multiple of 7.
The first 3 digit multiple of 7: 100 ÷ 7 = 14 2/7 → first 3 digit multiple of 7 is 15 × 7 = 105
Last 3 digit multiple of 7 less than 140: 140 ÷ 7 = 20 → last 3 digit multiple of 7 less than 140 is 19 × 7 = 133
→ The possible multiples of 7 to consider are [15..19] × 7 which gives 105, 112, 119, 126 and 133.
The sum of the digits of these number is:
105 → 1 + 0 + 5 = 6
112 → 1 + 1 + 2 = 4
119 → 1 + 1 + 9 = 11
126 → 1 + 2 + 6 = 9
133 → 1 + 3 + 3 = 7
Of these sums only 6, 4 and 7 are less than 9, thus the possible numbers are 105, 112 or 113.
