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
(4x³-12x +11) + (2x - 2)
Answers
4x^3-10x+9
Explanation
4x^3-12x+11+2x-2
Steve weatherspoon, a super salesman contemplating retirement on his 50th birthday, decides to create a fund on an 11% basis that will enable him to withdraw 14,570 per year on June 30th, beginning in 2024 and continuing through 2027. To develop this fund, Steven intends to make equal contributions on June 30th of each of the years 2020 through 2023. How much must the balance of the fund equal on June 30th 2023 in order for Steve to satisfy his objective?
Answers
Based on the amount that Steve Weatherspoon wants to withdraw every year beginning in June 30, 2024, and the interest rate, the balance on June 30th 2023 should be $45,203.
What should the balance be in 2023?The fact that Steve Weatherspoon wants to be able to withdraw a particular amount every year, this makes this amount an annuity.
The value in 2023 would therefore be the present value of the annuity that will then accrue to the required amounts as the years go by.
The present value of an annuity is:
= Annuity amount per year x Present value interest factor of an annuity, 11%, 3 years between 2024 and 2027
Solving gives:
= 13,126.25 x 3.44371
= $45,203
In conclusion, the balance on the fund in 2023 should be $45,203 in order for Steve Weatherspoon to achieve his objectives.
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Evaluate (f. g)(x) if f(x) = 2x² and g(x) = 3x - 2
Answers
Answer:
18x2 -24 x + 8
Step-by-step explanation:
( f o g) (x) = f ( g(x) ) = f( 3x-2) = 2( 3x-2)2 = 2( 9x2 - 12x + 4) = 18x2 -24 x + 8
Cai says you can divide both quantities in a ratio by the same nonzero number to
find an equivalent ratio. Explain why Cai is correct.(MATH)
Answers
Answer:
correct
Step-by-step explanation:
dividing both quantities in a ratio by the same non zero number does indeed find an equivalent ratio.
for example
60 : 36 ( divide both parts by 6 )
= 10 : 6 ← equivalent ratio ( divide both parts by 2 )
= 5 : 3 ← equivalent ratio in simplest form
in fact dividing the original by the HCF of the 2 quantities gives the ratio in simplest form immediately , that is
60 : 36 ( divide both parts by 12 )
= 5 : 3
Please help fast please
Answers
Answer:
See image
Step-by-step explanation:
It cant be wrong
Two trains leave towns 664 miles apart at the same time and travel toward each other. One train travels 16 mih faster than the other. If they meet in 4 hours, what is the rate of each train?
Answers
If the trains meet in 4 hours then the speed of the trains is 150 miles per hour and the speed of second train be 166 miles per hour.
Given the distance between trains be 664 miles and the speed of one train is 16 miles more than the other train.
We are required to find the speed of both the trains.
Speed is basically the distance that a thing covers in a particular time period.
Speed=Distance/Time
let the speed of first train be x miles per hour.
According to question the speed of the second train be (x+16) miles per hour.
Taking first train:
Speed =Distance/Time
x=664/4
=166
Taking second train:
Speed=Distance/Time
x+16=664/4
x+16=166
x=150
Hence if the trains meet in 4 hours then the speed of the trains is 150 miles per hour and the speed of second train be 166 miles per hour.
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a swimming pool charges £3.60 for entry. you can save 1/3 of entry fee with a membership card. on his first visit, ken spends £5 on a membership card plus the reduced entry fee. how many times does ken visit before he gets back his £5
Answers
The number of times Ken visits, given the £3.6 pool charge, and 1/3 savings per visit is 5 times.
Which method can be used to find the number of times Ken visits the pool to get back his £5?The entry fee = £3.60
Amount saved on entry fee by having a membership card = 1/3 of the entry fee
Amount Ken spends on the membership card and the reduced entry fee = £5
Therefore;
Reduced entry fee = £3.60×(1 - 1/3) = £2.4
Amount saved per visit = £3.6 - £2.4 = £1.2
The number of visits, n, before he gets back his £5 is therefore;
n = £5/(£1.2/visit) ≈ 4.17 visitsKen has to visit the swimming pool more than 4 times to get his £5 back.
Rounding to the next larger whole, therefore;
The number of visits, n, before he gets back his £5 is 5 visitsLearn more about fractions here:
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What is the probability that the top-three finishers in the contest will
all be seniors?
Type in the correct answer in each box. Use numerals instead of
words. If necessary, round your answers to the nearest tenth.
Answers
Answer: Just guessing, try.. or dont.
Step-by-step explanation:
There are 3 seniors. To make up 100% we would have to do 33.33%. I would say that the probabilty is around 33.33, and don't know the different orders possible.
Give that m/_ y=39, find the measures of angles a and b
Answers
Answer:
∠a = 39°
∠b = 129°
Step-by-step explanation:
Vertical angles are congruent. An exterior angle of a triangle is equal to the sum of the remote interior angles.
ApplicationAngle y and angle 'a' are vertical angles, so congruent.
∠a = ∠y = 39°
The angle marked 'b' is an exterior angle to the triangle. Its remote interior angles are 'a' and the one marked 90°.
∠b = ∠a +90° = 39° +90°
∠b = 129°
[tex] \huge\mathbb{ \underline{SOLUTION :}}[/tex]
Given:[tex]\longrightarrow\bold{m<a= \: 39}[/tex] ✓[tex]\longrightarrow\bold{m<b= \: 141}[/tex]In the given figure, angles a and y are vertically opposite angles. The measure of vertical angles is equal, therefore :-
[tex]\small\longrightarrow\sf{m<a = m<y}[/tex]
[tex]\small\longrightarrow\sf{m<a= \: 39^\circ}[/tex]
Next, angles a and 90° are the opposite interior angles to the exterior angle b. By the triangle theorem below :-
[tex]\small\longrightarrow\sf{m<b=m<a+90^\circ}[/tex]
[tex]\small\longrightarrow\sf{39^\circ+90^\circ}[/tex]
• [tex]\small\longrightarrow\sf{m<b= 129^\circ}[/tex]
[tex]\huge \mathbb{ \underline{ANSWER:}}[/tex]
m<a= [tex]\large\sf{\boxed{\sf 39^\circ}}[/tex]
m<b = [tex]\large\sf{\boxed{\sf 129^\circ}}[/tex]
Inga is solving 2x2 + 12x – 3 = 0. Which steps could she use to solve the quadratic equation? Select three options. 2(x2 + 6x + 9) = 3 + 18 2(x2 + 6x) = –3 2(x2 + 6x) = 3 x + 3 = Plus or minus StartRoot StartFraction 21 Over 2 EndFraction EndRoot 2(x2 + 6x + 9) = –3 + 9
Answers
The required steps are
2x^2 + 12x – 3 = 0.
2x2 + 12x = 3
2(x^2+6x) = 3
2[(x+3)^2 - 9] = 3
Completing the square methodQuadratic equations are equations that has a leading degree of 2
Given the equation below;
2x^2 + 12x – 3 = 0.
Add 3 to both sides
2x2 + 12x = 3
Factor out 2
2(x^2+6x) = 3
Complete the square
2(x^2+6x+3^2-3^2) = 3
2[(x+3)^2 - 9] = 3
This gives the required steps to solve the equation
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Help me with this I need help asap!
Answers
Answer:
29.5
Step-by-step explanation:
[tex]UV=\frac{27+32}{2} \\ \\ UV=29.5[/tex]
Ashlee sells lunches for $7.00 and drinks for $2.25. One busy Saturday, Ashlee sold $440.00 worth of lunches and drinks. Which equation represents the relation between the number of lunches sold, x, the number of drinks sold, y, and the amount in dollars that Ashlee sold that day?
Group of answer choices
Answers
Answer:
7x + 2.25y = 440
Step-by-step explanation:
Ashlee sells lunches for $7.00 and drinks for $2.25.
One busy Saturday, Ashlee sold $440.00 worth of lunches and drinks.
The number of lunches sold is x.
The number of drinks sold is y.
The appropriate equation will be:
= (7 × x) + (2.25 × y)
7x + 2.25y = 440
Which of the graphs below shows the solution set of -7 < x <0
Answers
Answer:
none
Step-by-step explanation:
A. -7 <= x < 0
B. x <= -7 or x > 0
C. -7 < x <= 0
D. x < -7 or x >= 0
To create a graph to represent -7 < x < 0, draw empty circles at -7 and 0, then connect the two with a line.
Vanessa knows that about 20% of the students in her school are bilingual. She wants to estimate the probability that if 3 students were randomly selected, all 3 of them would be bilingual. To do so, she will use a spinner with 5 equally sized sections labeled as either "bilingual" or "monolingual". She will spin the spinner 3 times, record the results, and repeat the process 50 times.
How many of the 5 sections should she label "bilingual"?
How many of the 5 sections should she label "monolingual"?
Answers
Answer:
1 section should be labelled "bilingual".
4 sections should be labelled "monolingual".
Step-by-step explanation:
If 20% of students in the school are bilingual, then 80% of students in the school are monolingual, since 100% - 20% = 80%.
If the spinner is to be divided into 5 equally sized sections, with each section labeled as "bilingual" or "monolingual" to represent the corresponding proportion of students in the school, then:
Bilingual
20% of 5
= 20/100 × 5
= 100/100
= 1 section
Monolingual
80% of 5
= 80/100 × 5
= 400/100
= 4 sections
Therefore:
1 section should be labelled "bilingual".4 sections should be labelled "monolingual".What is the measure of the third angle?
Answers
Answer:
108
Step-by-step explanation:
11+61=72
We use 180 and subtract it by 72 because we know that all angles in a triangle add up to 180.
180-72=108
The double number line shows that the distance from the start of a hiking trail to a cave
entrance is 700 m.
Answers
Answer:
100, 25, 50
Step-by-step explanation:
[tex]700: \frac{700}{700} \times 100=100 \\ \\ 175: \frac{175}{700} \times 100=25 \\ \\ 350: \frac{350}{700} \times 100=50[/tex]
4a*(3-2b) for a=1/2 and b=-1
pls hurry
Answers
Answer:
2
Step-by-step explanation:
Use pemdas
4a*(3-2b)
4(1/2)*(3-2(-1))
4* 1/2 = 2
2*(3-2(-1)
-2*-1 = 2
2(3-2)
3 -2 = 1
2(1) = 2
Hello guys, can you please help my son!!
Answers
Based on the fact that there is no country which had won more than 900 silver medals, set E using the roster method would be E = { }.
How is set E denoted?Set E is meant to include the set of countries who have attained more than 900 silver medals in the Olympic games.
The highest number of silver medals attained is that of the United States which is 860 silver medals.
This means that no single country has attained more than 900 silver medals.
Set E would therefore be null. Using the roster method, the way to show set E would be:
E = { }
In conclusion, set E is E = { }.
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Which polygon does not belong with the others?
Answers
This is because all of the other choices are equal sided, which makes them a regular polygon.
Can u guys please give me the correct answe
Answers
Answer:
35
Step-by-step explanation:
Since the two sides are equal in a parallelogram angle B = y ,so we have angle to be equal to 145°
We sum both angles and divide from
360° - [145° +145°]
360° - 290° = 70
Since the left sides( X And Z) are equal then we divide what we got into 2
70 ÷ 2 = 35
therefore angle X equals 35
Manufacturers of tires report that car tires should be able to last an average of 50,000 miles. A new tire company produces a different type of tread and tests 100 randomly selected tires. This sample of 100 tires lasted an average of 51,500 miles. Assuming the new type of tread does not improve the mileage of the tire, 200 sample means were simulated and displayed on the dotplot.
Using the dotplot and the sample mean mileage, is there convincing evidence that the new type of tread improves the mileage of the tire?
Yes, because a mean mileage of 51,500 or more occurred only 14 out of 200 times, the mean mileage is statistically significant. There is convincing evidence the new type of tire tread improves mileage of the tire.
Yes, because a mean mileage of 51,500 or less occurred 186 out of 200 times, the mean mileage is statistically significant. There is convincing evidence the new type of tire tread improves mileage of the tire.
No, because a mean mileage of 51,500 or less occurred 186 out of 200 times, the mean mileage is not statistically significant. There is not convincing evidence the new type of tire tread improves mileage of the tire.
No, because a mean mileage of 51,500 or more occurred 14 out of 200 times, the mean mileage is not statistically significant. There is not convincing evidence the new type of tire tread improves mileage of the tire.
Answers
There is convincing evidence the new type of tire tread improves mileage of the tire.
How to interpret the mean mileage?The given parameter is:
Average = 50000 miles
New average = 51500 miles
The dot plot is added as an attachment
From the dot plot, we can see that the mean value of 51500 or more occurred 14 times out of the 200 samples
So, the conclusion is that: (a) Yes, because a mean mileage of 51,500 or more occurred only 14 out of 200 times, the mean mileage is statistically significant. There is convincing evidence the new type of tire tread improves mileage of the tire.
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Find the length of the a of a right triangle b=21.5 inches and the hypotenuse c =31.9.use the calculator to estimate the square root to one decimal place
The length of the leg a is approximately?
Answers
Answer:
Step-by-step explanation:
a² + b² = c²
a² + (21.5)² = (31.9)²
a² + 462.25 = 1017.61
a² = 555.36
a = 23.6
Screenshot below will hold the question.
Answers
The equation the completes the proof in step 6 is m<w + m<x = m<x + m<z
How to complete the proof?In step 4 of the proof, we have the following equation
m<w + m<x = m<y + m<z
In step 5, we have
m<x = m<y
When m<x = m<y is substituted to the equation in step 4, we have:
m<w + m<x = m<x + m<z
Hence, the equation the completes the proof in step 6 is m<w + m<x = m<x + m<z
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If the next two Junior Athletics events are sold out, the new table will look like this: Event 1 Event 2 Event 3 Event 4 Event 5 Event 6 Event 7 Junior Athletics Sold Out Sold Out Not Sold Out Not Sold Out Sold Out Sold Out Sold Out What is the new probability of the event being sold out? Give your answer as a fraction.
Answers
0.7 is the new probability of the event being sold out given that total number of events is 7 and 5 events are sold out. This can be obtained by using the formula for probability.
Find the new probability of the event being sold out:Probability is the chance of occurrence of an event.
⇒ The formula for finding probability,
Probability = [tex]\frac{Number\ of\ favourable\ outcomes}{Total\ number\ of\ outcomes}[/tex]
Here it is given in the question that,
Total number of outcomes = Total number of events = 7 Number of events sold out = 5 Number of events not sold out = 2Therefore by using the formula of probability we get,
⇒ Probability (Junior Athletics being sold out) = [tex]\frac{Number\ of\ favourable\ outcomes}{Total\ number\ of\ outcomes}[/tex]
Probability (Junior Athletics being sold out) = [tex]\frac{Number\ of\ events\ sold\ out}{Total\ number\ of\ events}[/tex]
Probability (Junior Athletics being sold out) = 5/7
⇒ Probability (Junior Athletics being sold out) = 0.7
Hence 0.7 is the new probability of the event being sold out given that total number of events is 7 and 5 events are sold out.
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A pack of soil weighs 43 lbs. Each plant pot requires just 12 lbs of soil.
Calculate how many plant pots can be filled.
Answers
If we divide 43 lbs into pots of 12 lbs, we end up getting 3 pots with a remainder (what is left) of 7 lbs. However, in a scenario such as this, we discard (ignore, get rid of) the remainder since it is not enough to completely fill a plant pot.
A fish can swim at 14m/s with the current and at 6m/s against it. Find the speed
of the current and the speed of the fish in still water.
Answers
Answer:
10 m/s
Step-by-step explanation:
Let x = speed of fish and y = speed of current. When swimming with current, the fish's speed is x + y. When swimming against it, the speed is x - y. Therefore, we can set up a system of linear equations to solve for x, the speed of fish.
x + y = 14
x - y = 6
2y = 8
y = 4
x - 4 = 6
x = 10
The speed of fish in still water is 10 m/s, and the speed of current is 4 m/s.
Answer:
Speed of fish in still water = 10 m/s
Speed of current = 4 m/s
Step-by-step explanation:
Let the speed of the fish in still water = x m\s
Let the speed of the current = y m/s
Speed of the fish with the current = x + y
x +y = 14 m/s ------------------(I)
Speed of the fish against current = x -y
x - y = 6 m/s ----(II) &
Add the equations (I) & (II) and 'y' will be eliminated and we can find the value of 'x'.
x + y = 14
x - y = 6 {Now, add)
2x = 20
x = 20/2
x = 10
[tex]\sf \boxed{\text{\bf Speed of fish in still water = 10 \ m / s}}[/tex]
Plugin x = 10 in equation (I)
10 + y = 14
y = 14 - 10
y = 4
[tex]\sf \boxed{\text{\bf Speed of the current = 4 \ m/s}}[/tex]
(-4,2) (5,0) midpoint
Answers
Answer:
[tex]M = (0.5, 1)[/tex]
Step-by-step explanation:
The midpoint can be found using the formula: [tex]x, y = (\frac{x_2+x_1}{2}, \frac{y_2 + y_1}{2})[/tex]
So plugging in the values we get: [tex]M = (\frac{-4+5}{2}, \frac{2+0}{2})\\M = (\frac{1}{2}, \frac{2}{2})\\M = (0.5, 1)[/tex]
Answer: [tex]\Large\boxed{Midpoint~=~(\frac{1}{2} ,~1)}[/tex]
Step-by-step explanation:
Given information
(x₁, y₁) = (-4, 2)
(x₂, y₂) = (5, 0)
Given formula for midpoint
[tex]Midpoint~=~(\frac{x_1~+~x_2}{2} ,~\frac{y_1~+~y_2}{2} )[/tex]
Substitute values into the formula
[tex]Midpoint~=~(\frac{-4~+~5}{2} ,~\frac{2~+~0}{2} )[/tex]
Simplify by addition
[tex]Midpoint~=~(\frac{1}{2} ,~\frac{2}{2} )[/tex]
Simplify fractions if necessary
[tex]\Large\boxed{Midpoint~=~(\frac{1}{2} ,~1)}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Evaluate this exponential expression.
OA. 19
OB. 63
OC. 66
OD. 207
6 (4+2)²-32
Answers
Answer: 184
(I am not sure if I have misunderstood the question, but the final answer I got was 184, which wasn't any one of the choices. I was wrong in understanding, please let me know and I will modify my answer)
Step-by-step explanation:
Given expression
6 (4 + 2)² - 32
Simplify values in the parenthesis
6 (6)² - 32
Simplify exponents
6 × 36 - 32
Simplify multiplication
216 - 32
Simplify subtraction
[tex]\Large\boxed{184}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
A certain cellular phone plan charges $31.00 per month plus $0.14 for each minute of usage. The monthly charge is given by the formula monthly charge=0.14x+31, where x represents the number of minutes of usage per month. What is the charge for a month with 340 minutes of usage? Write your answer to two decimal places as money is traditionally written.
Answers
Answer:
$78.60
Step-by-step explanation:
0.14x + 31 = 0.14*340 + 31 = 78.6
Carlos went to PetSmart to get his hamster an exercise ball. There was a small one with a 5 inch diameter and a 13 inch ball. How much more space (volume) is in the larger than the smaller? Round to the nearest tenth.
Answers
Answer:
150.8 in³
Step-by-step explanation:
The volume of the 5 inch ball is
[tex]\frac{4}{3}\pi (5/2)^{3}[/tex]
The volume of the 13 inch ball is
[tex]\frac{4}{3}\pi (13/2)^{3}[/tex]
Subtracting these volumes, we get about 150.8 in³.
The amount of space (volume) that is in the larger spherical ball than the smaller will be 1,084.8 in.³
What is the Volume of a Spherical Solid?To find the volume of a sphere, which is the amount of space the solid contains, the formula used is given as: 4/3 πr³,
where r is the radius of the sphere, which is half the measure of the diameter.
Diameter of the smaller spherical ball = 5 inches
Radius = 5/2 = 2.5 inches
Volume of the smaller spherical ball = 4/3 πr³ = 4/3 × π × 2.5³
Volume ≈ 65.5 in.³
Diameter of the larger spherical ball = 13 inches
Radius = 13/2 = 6.5 inches
Volume of the larger spherical ball = 4/3 πr³ = 4/3 × π × 6.5³
Volume of the larger spherical ball ≈ 1,150.3 in.³
The amount of space (volume) that is in the larger spherical ball than the smaller.
= 1,150.3 - 65.5
The amount of space (volume) that is in the larger spherical ball than the smaller will be 1,084.8 in.³
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