Looks like your function is
[tex]f(x)=\dfrac x{6x^2+1}[/tex]
Rewrite it as
[tex]f(x)=\dfrac x{1-(-6x^2)}[/tex]
Recall that for [tex]|x|<1[/tex], we have
[tex]\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n[/tex]
If we replace [tex]x[/tex] with [tex]-6x^2[/tex], we get
[tex]f(x)=\displaystyle x\sum_{n=0}^\infty\frac(-6x^2)^n=\sum_{n=0}^\infty (-6)^n x^{2n+1}[/tex]
By the ratio test, the series converges if
[tex]\displaystyle\lim_{n\to\infty}\left|\frac{(-6)^{n+1} x^{2(n+1)+1}}{(-6)^n x^{2n+1}}\right|=6|x^2|\lim_{n\to\infty}1=6|x|^2<1[/tex]
Solving for [tex]x[/tex] gives the interval of convergence,
[tex]|x|^2<\dfrac16\implies|x|<\dfrac1{\sqrt6}\implies -\dfrac1{\sqrt 6}<x<\dfrac1{\sqrt 6}[/tex]
We can confirm that the interval is open by checking for convergence at the endpoints; we'd find that the resulting series diverge.
The interval of the convergence is (-1/√6 < x < 1/√6). We can confirm that the interval is open by checking for convergence at the endpoints.
What is a function?The function is an expression, rule, or law that defines the relationship between one variable to another variable. Functions are ubiquitous in mathematics and are essential for formulating physical relationships.
The given function is
[tex]\rm f(x) = \dfrac{x}{6x^2 + 1} \\\\or \\\\f(x) = \dfrac{x}{1 - (-6x^2)}[/tex]
For |x| < 1, we have
[tex]\rm \dfrac{1}{1-x} = \Sigma_{n=0}^{\infty} \ x^n[/tex]
If x is replaced with -6x², then we have
[tex]\rm f(x)= \Sigma_{n=0}^{\infty} (-6x^2 )^n = \Sigma_{n=0}^{\infty} (-6)^n x^{2n+1}[/tex]
Then by the ratio test, the series converges if
[tex]\displaystyle \lim_{n \to \infty} \left| \dfrac{(-6)^{n+1}x^{2(n+1)+1}}{(-6)^{n}x^{2n+1}} \right|=6|x^{2}| \displaystyle \lim_{n \to \infty }1=6|x^{2}| < 1[/tex]
Solving for x, the interval of convergence will be
[tex]|x^2| < \dfrac{1}{6} \\\\|x| < \dfrac{1}{\sqrt6} \\\\-\dfrac{1}{\sqrt6} < x < \dfrac{1}{\sqrt6}[/tex]
We can confirm that the interval is open by checking for convergence at the endpoints.
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Find the area of the shaded figure.
Answer:
8500 ft²
Step-by-step explanation:
First we need to find the area of the non shaded region within this shaded region. To find the area of this non shaded rectangle, use the formula base×height.
20×100=2000 ft².
Now we need to find the height and width of the shaded region which is,
25+25+20=70 Height
25+25+100=150 Width
mutiply these numbers 150×70=10500 ft²
Now we just need to subtract that inside area.
10500-2000=8500 ft²
If the square root of n is approximately equal to 4.2, then n is between _____.
4 and 5
16 and 25
8 and 9
2 and 3
Answer:
16 and 25
Step-by-step explanation:
Given that the square root of n = 4.2, let's find an expression for this, then find the possible value of n.
Thus:
[tex] \sqrt{n} = 4.2 [/tex]
Solve for n. Square both sides.
[tex] (\sqrt{n})^2 = (4.2)^2 [/tex]
[tex] n = 17.64 [/tex]
Therefore, if the sqrae root of n was approximately equal to 4.2, then we can conclude that n is between 16 and 25.
Answer:
16 and 25
Step-by-step explanation:
4. Both the Galapagos Islands and the island of Naura are on the Equator, but the Galapagos Islands are at 90.30◦W whereas the island of
Nauru is at 166.56◦E. How far is it from the Galapagos Islands to
Nauru traveling over the Pacific ocean along the Equator, correct to
the nearest km ?
Answer:
Distance between Nauru and Galapagos islands = 11525 Km to the nearest Km
Step-by-step explanation:
The angle between the two Islands is given as X
X = (180 - 166.56) + (180 - 90.30)
X = 13.44° + 89.70°
X = 103.14
Distance between the islands = length of the arc with angle X subtended at the center of the earth of radius R.
Length of arc = (X/360) × 2πR
Where, R, radius of the earth = 6400 Km
Length of arc = (103.14/360) × 2π × 6400 Km
Length of arc = 11525.48 Km
Therefore, distance between Nauru and Galapagos islands = 11525 Km to the nearest Km
Amy invested $300 in the bank and a year later has $343.70. By what percent has the amount changed?
44% increase
13% increase
15% increase
87% increase
Answer:
15
Step-by-step explanation:
i just know
Answer:
15
Step-by-step explanation:
just took the quiz
Find the area of the surface. The part of the plane x + 2y + 3z = 1 that lies inside the cylinder x2 + y2 = 4
The area of a surface between a plane and a cylinder is evaluated using the integral [tex]\int\limits \int\limits^{}_D {\sqrt{(\frac{dz}{dx})^2 + (\frac{dz}{dy})^2 + 1} } \, dA[/tex]. So, the area of the given surface is [tex]\frac{4\pi}{3}\sqrt{14}[/tex]
Given that
[tex]x + 2y + 3z =1[/tex]
[tex]x^2 + y^2 = 4[/tex]
Make z the subject in [tex]x + 2y + 3z =1[/tex]
[tex]3z = 1 - x - 2y[/tex]
[tex]z = \frac{1}{3}(1 - x - 2y)[/tex]
The surface area is calculated using the formula:
[tex]Area = \int\limits \int\limits^{}_D {\sqrt{(\frac{dz}{dx})^2 + (\frac{dz}{dy})^2 + 1} } \, dA[/tex]
Where: [tex]dA = rdr \times d\theta[/tex]
[tex]z = \frac{1}{3}(1 - x - 2y)[/tex]
Calculate [tex]\frac{dz}{dx}[/tex]
[tex]\frac{dz}{dx}= \frac{1}{3}(0 - 1 - 2 \times 0)[/tex]
[tex]\frac{dz}{dx}= \frac{1}{3}(0 - 1 - 0)[/tex]
[tex]\frac{dz}{dx}= \frac{1}{3}(- 1)[/tex]
[tex]\frac{dz}{dx}= -\frac{1}{3}[/tex]
Calculate [tex]\frac{dz}{dy}[/tex]
[tex]\frac{dz}{dy}= \frac{1}{3}(0 - 0 - 2 \times 1)[/tex]
[tex]\frac{dz}{dy}= \frac{1}{3}(- 2)[/tex]
[tex]\frac{dz}{dy}= -\frac{2}{3}[/tex]
Because the plane is inside [tex]x^2 + y^2 = 4[/tex], then the region of z is:
[tex]D = \{(r,\theta) | 0 \le r \le \sqrt{4}, 0 \le \theta \le 2\theta\}[/tex]
[tex]D = \{(r,\theta) | 0 \le r \le 2, 0 \le \theta \le 2\theta\}[/tex]
[tex]Area = \int\limits \int\limits^{}_D {\sqrt{(\frac{dz}{dx})^2 + (\frac{dz}{dy})^2 + 1} } \, dA[/tex] becomes
[tex]Area = \int\limits^{2\pi}_0 \int\limits^{2}_0 {\sqrt{(\frac{-1}{3})^2 + (\frac{-2}{3})^2 + 1} } \, dA[/tex]
[tex]Area = \int\limits^{2\pi}_0 \int\limits^{2}_0 {\sqrt{\frac{1}{9} + \frac{4}{9} + 1} } \, dA[/tex]
Take LCM
[tex]Area = \int\limits^{2\pi}_0 \int\limits^{2}_0 {\sqrt{\frac{1+4+9}{9}} } \, dA[/tex]
[tex]Area = \int\limits^{2\pi}_0 \int\limits^{2}_0 {\sqrt{\frac{14}{9}} } \, dA[/tex]
Evaluate the square root of 9
[tex]Area = \int\limits^{2\pi}_0 \int\limits^{2}_0 {\frac{\sqrt{14}}{3} } \, dA[/tex]
Remove the constant
[tex]Area = \frac{\sqrt{14}}{3}\int\limits^{2\pi}_0 \int\limits^{2}_0 { dA}[/tex]
Recall that: [tex]dA = rdr \times d\theta[/tex]
So, we have:
[tex]Area = \frac{\sqrt{14}}{3}\int\limits^{2\pi}_0 \int\limits^{2}_0 { rdr \times d\theta}[/tex]
Integrate
[tex]Area = \frac{\sqrt{14}}{3}\int\limits^{2\pi}_0 { \frac{r^2}{2} |\limits^{2}_0 \ d\theta}[/tex]
Expand
[tex]Area = \frac{\sqrt{14}}{3}\int\limits^{2\pi}_0 { \frac{2^2 - 0^2}{2} \ d\theta}[/tex]
[tex]Area = \frac{\sqrt{14}}{3}\int\limits^{2\pi}_0 { \frac{4}{2} \ d\theta}[/tex]
[tex]Area = \frac{\sqrt{14}}{3}\int\limits^{2\pi}_0 { 2 \ d\theta}[/tex]
Integrate
[tex]Area = \frac{\sqrt{14}}{3}[2\pi]|\limits^{2\pi}_0[/tex]
Expand
[tex]Area = \frac{\sqrt{14}}{3}[2 \times 2\pi - 0][/tex]
[tex]Area = \frac{\sqrt{14}}{3}[4\pi - 0][/tex]
[tex]Area = \frac{\sqrt{14}}{3}[4\pi][/tex]
[tex]Area = \frac{4\pi}{3}\sqrt{14}[/tex]
Hence, the area of the surface is: [tex]\frac{4\pi}{3}\sqrt{14}[/tex]
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A man buys a racehorse for $20,000 and enters it in two races. He plans to sell the horse afterward, hoping to make a profit. If the horse wins both races, its value will jump to $90,000. If it wins one of the races, it will be worth $55,000. If it loses both races, it will be worth only $15,000. The man believes there is a 30% chance that the horse will win the first race and a 40% chance that it will win the second one. Assuming that the two races are independent events, find the man's expected profit.
The expected profit of the man from the horse race is $22,400.
Given data:
To find the man's expected profit, calculate the probability of each outcome and multiply it by the corresponding profit.
To determine the probability of each outcome based on the given information:
Probability of winning both races = 0.30 * 0.40
Probability of winning both races = 0.12
Probability of winning one race = 0.30 * 0.60 + 0.70 * 0.40
Probability of winning one race = 0.46
Probability of losing both races = 0.70 * 0.60
Probability of losing both races = 0.42
Next, calculate the expected profit for each outcome:
Profit from winning both races = $90,000 - $20,000
Profit from winning both races = $70,000
Profit from winning one race = $55,000 - $20,000
Profit from winning one race = $35,000
Profit from losing both races = $15,000 - $20,000
Profit from losing both races = -$5,000 (a loss of $5,000)
Now, determine the man's expected profit by multiplying the probability of each outcome by the corresponding profit and summing them up:
Expected profit = (0.12 * $70,000) + (0.46 * $35,000) + (0.42 * -$5,000)
Expected profit = $8,400 + $16,100 - $2,100
Expected profit = $22,400
Hence, the man's expected profit is $22,400
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Which expression is the exponential form of
Answer:
it would be the second one cause i picked it and it was right
Step-by-step explanation:
A rectangular Persian carpet has a perimeter of 220 inches the length of the carpet is 22 inches more than the width what are the dimensions of the carpet
Answer:
length= 65 inches
width= 45 inches
Step-by-step explanation:
65 plus 65= 130
45 plus 45= 90
130 plus 90= 220 in.
Graph the line with the slope -3/4 passing through the point (4,-1)
Answer:
Step-by-step explanation:
Plot the point (4, -1) in the 4th Quadrant.
The slope (-3/4) tells us what to do next:
With your pencil point initially on (4, -1), move it 4 units to the right, to (8, -1).
With your point initially on (8, -10, move it down 3 units, to (8, -4). Plot (darken) (8, -4).
Draw a line through (4, -1) AND (8, -4)
The equation of a line for the given slope and the coordinate point is 4y=-3x+8.
Given that, the line with the slope -3/4 passes through the point (4,-1).
We need to find the line of equations.
What is slope intercept form?The slope-intercept form of a straight line is used to find the equation of a line. For the slope-intercept formula, we have to know the slope of the line and the intercept cut by the line with the y-axis. Let us consider a straight line of slope 'm' and y-intercept 'b'. The slope intercept form equation for a straight line with a slope, 'm', and 'b' as the y-intercept can be given as y = mx + b.
Now, substitute m=-3/4 and the point (4,-1) in y = mx + b and find the value of b.
That is, -1 = -3/4 (4)+b
⇒b=2
Then the equation becomes y=-3/4 x+2
⇒4y=-3x+8
Therefore, the equation of a line for the given slope and the coordinate point is 4y=-3x+8.
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Point W has coordinates (4, -7). If it is reflected over the y-axis. What are the coordinates of its image? (4, -7) (-4, 7) (-4, -7) (4, -7)
Answer:
(-4, -7)
Step-by-step explanation:
reflection over the y-axis means the x-value becomes the opposite (+ → - or - → +)
(4, -7) → (-4, -7)
Answer:
(-4, -7)
Step-by-step explanation:
Ella purchase 350 pallets of flowers there are 152 flowers on each pallet how many flowers did Ella purchase?
Answer:
53,200
Step-by-step explanation:
350 pallets times 152 flowers each is 53,200 flowers in total
Please give brainliest thanks
Answer:
53,200
Step-by-step explanation:
I saw the other guys answer
The following is a list of 5 measurements. 19,30,20,10,17 Suppose that these 5 measurements are respectively labeled.
Answer:
For example, suppose we weigh five children. ... 5. ∑ i=1 xi. = x1 + x2 + x3 + x4 + x5. = 10 + 12 + 14 + 8 + 11. = 55. ... We also use sigma notation in the following way: 4 ... j2 = 12 + 22 + 32 + 42 = 30.
Missing: 17 | Must include: 17Summary measures for this data set are ... 17 18. 19 20. Observedresult. 0. 1. 0. 1. 1. 0. 1. 0. 1. 1. Total so far. 4. 5. 5. 6 ... 5. 10. 15. 20. 25. 30. 35. 40 . Toss. Figure S2.1 Proportion P, 40 tosses of a coin.
Series is defined as sum of sequences. The sum of the sequence given is 96
Sequence and seriesSeries is defined as sum of sequences
Given the sequence below
19,30,20,10,17
The given su notation denotes the sum of the first five terms of the sequence.
Take the sum
Sum = 19+30+20+10+17
Sum = 96
Hence the sum of the sequence given is 96
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What is 400,000 in scientific notation?
Answer:
Hello! I hope I am correct!
Step-by-step explanation:
Answer:
4x 10^5
Make it into decimal
Decimal form:4.00000
Now count the zeros in 4.00000
There is 5 zeros and 4 (4 is the first number)
Formula: Ax10^{B}
Answer: 4x10^{5}
That’s how you get your answer.
Hope this helps!
By: BrainlyAnime
Rita bought 6 boxes of chocolates. Each box cost $6.48. How much did she spend?
Answer:
38.88
Step-by-step explanation:
Take the price per box and multiply by the number of boxes
6.48*6 =38.88
Between which two positive integers does ;33 lie?
Listed below are the budgets (in millions of dollars) and the gross receipts (in millions of dollars) for randomly selected movies. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using alpha = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between budgets and gross receipts? Do the results change if the actual budgets listed are $61,000,000, $92,000,000, $48,000,000, and so onBudget(x) 61 92 48 36 135 58 93Gross(y) 69 62 53 58 627 144 42a. What are the null and alternative hypotheses?i) H_0: rho = 0, H_1: rho < 0ii) H_0: rho = 0, H_1: rho notequalto 0iii) H_0: rho notequalto 0, H_1: rho = 0iv) H_0: rho = 0, H_1: rho > 0b. Construct a scatterplot. Choose the correct graph below.c. The linear correlation coefficient r is.d. The test statistic t is.e. The P-value is.
Answer:
The answers are below
Step-by-step explanation:
1. H0: rho = 0
H1: rho not equal to 0
2. I will add the scatter plot as an attachment
3. ΣX = 523,
ΣY = 1055
ΣX² = 46023
ΣY²= 430407
ΣXY = 111448
n = 7
The correlation coefficient r =
r = 7(111448)-(523)(1055)/√7(46023)-(523)² × √7(430407)-(1055)²
r = 780136-551765/√322161-273529√3012849-1113025
= 228371/220.53*1378.34
= 228371/303965.32
r = 0.751
4. Test statistics
t = r√n-2/√1-r²
= 0.751√5/√0.5640
= 1.679/0.660
= 2.543
5. P value
Degrees of freedom n-2 = 5
T dist(2.543,5,2)
The p value is less than 0.05 so we reject null hypothesis. There is enough evidence to show a linear correlation between budgets and gross receipts.
A sports conference has 13 teams. It was proposed that each team play precisely one game against each of exactly seven other conference teams. Prove that this proposal is impossible to implement.
Step-by-step explanation:
Number of team = 13
Let us consider a graph with 13 vertices, each edge represents a match of two teams.
Total number of edges = 13(13-1)/2 =78 edge
Given conditions:
Each component plays 7 match with 7 other teams.
Total edges in the case = 7(6)/2 =21
Therefore edges in the graph are not equal to edges in the condition.
Therefore impossible to implement
1mm66 converted in height inches
Answer:
0.065354330709 inch
Step-by-step explanation:
1 millimeter is equal to 0.03937007874 inches, so 1.66 mm is 0.065354330709 inches.
This comes from the definition 1 inch = 2.54 cm
so 1 inch = 25.4 mm. Divide by 25.4 and get:
0.03937007874 inch = 1 mm
Which plane figure generates a cylinder when it rotates about the dashed line?
Answer:
It is a rectangle
Find the domain of the relation represented by the list of ordered pairs below (-2,9), (6,-7), (-9,-5), (-12,10)
Domain of the relation → {-12, -9, -2, 6}
Given in the question,
List of ordered pairs,
(-2, 9), (6, -7), (-9, -5), (-12, 10)In a ordered pair (h, k), domain is defined by the x-values and range is defined by the y-values.
Therefore, domain of the relation represented by the ordered pairs will be {-12, -9, -2, 6}
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-3(k-8)-(k+5)=23
please explain how to do this
A password contains three digits, such as 175. How many different passwords can be formed?
Answer:
157,517,715,751,571.Therefore,5 more passwords can be formed
ASAP what is 46x78+(56-43)247.Our teacher gave us random problems cuz we were horrible
Answer: 6,799
Step-by-step explanation:
Solve fora - 26 = - 2w Simplify your answer as much as possible
Answer:
w = 13
if you have any questions just ask!
Please Help I’ve been stuck on these questions for a while now!
Answer:
written below for problem #2
Step-by-step explanation:
a. ratio is 12:40 since they're are 12 non fiction and 40 graphic novels
b. 40:12 (same reasoning as part a)
c. 6:40 for the ratio of science fiction to graphic novel, 12:6 for the ration of non fiction to science fiction, x:6 for the ratio of the unknown amount of fiction books to the science fiction books. (x is just there as a variable so it just represents fiction books)
What is the decibel level of a dog barking with intensity 10−5 watts per square inch? Use a logarithmic model to solve.
Answer:
70 dB
Step-by-step explanation:
D=10log(I10−12)
Substitute in the intensity level, I, and then simplify to find
DD=10log(10−510−12)=10log(107).
Since log107=7, simplify to find
DD=10⋅7=70.
The decibel level is 70dB.
Determine the sample size required to estimate the mean score on a standardized test within 4 points of the true mean with 98% confidence. Assume that s = 14 based on earlier studies.
Answer: 67
Step-by-step explanation:
Formula to calculate sample size: [tex]n=(\dfrac{z^*\times s}{E})^2[/tex] , where s = standard deviation on prior studies , z* = two tailed critical value for confidence level , E = margin of error
Given: E = 4 , s= 14 and confidence level = 98%
Z-value for 98% confidence = 2.326
Then, required sample size: [tex]n=(\dfrac{2.326\times 14}{4})^2=(8.141)^2=66.275881\approx67[/tex]
Hence, the required sample size = 67 .
Sample size required to estimate the mean score on a standardized test is 67.
The confidence level (C) = 98% = 0.98
α = 1 - C = 0.02
α/2 = 0.02/2 = 0.01
The z score of α/2 is the same as the z score of 0.49 (0.50 - 0.01) which is equal to 2.33
Given that:
Standard deviation (σ) = 14, margin of error (E) = 4, n = sample size, hence:
[tex]E=z_\frac{\alpha}{2}*\frac{\sigma}{\sqrt{n} } \\\\4=2.33*\frac{14}{\sqrt{n} } \\\\\sqrt{n}=8.155\\\\n=67[/tex]
Hence the sample size required to estimate the mean score on a standardized test is 67.
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Find the mode. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The mode(s) is(are) nothing. (Type an integer or a decimal. Do not round. Use a comma to separate answers as needed.) B. There is no mode.
Answer:
The mode of the given data is 8.
Step-by-step explanation:
The complete question is: The number of credits being taken by a sample of 13 full-time college students are listed below. Find the mean, median, and mode of the data, if possible. If any measure cannot be found or does not represent the center of the data, explain why.
9, 9, 12, 12, 9, 10, 8, 8, 8, 8, 8, 8, 11.
Find the mode. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The mode(s) is/are nothing . (Type an integer or decimal rounded to one decimal place as needed. Use a comma to separate answers as needed.) B. The data set does not have a mode.
We are given the data for the number of credits being taken by a sample of 13 full-time college students below;
Credits: 9, 9, 12, 12, 9, 10, 8, 8, 8, 8, 8, 8, 11.
Mode tells us that which value in our data set is appearing or occurring the maximum number of times.
As we can see in our data that the occurrence of the following credits are;
9 is occurring 3 times in our data.
12 is occurring 2 times in our data.
10 is occurring 1 time in our data.
8 is occurring 6 times in our data.
11 is occurring 1 time in our data.
Here, it is clear that the credit value of 8 is appearing the maximum number of times in our data set. So, the mode of the given data is 8.
Currently, Nora is 3 times as old as Damon. In 8 years, she will only be 2 times as old as Damon. How old will Damon be in 8 years?
Answer:
The answer is 16
Step-by-step explanation:
How many is (k)/(4)=(3)/(8)
Answer:
k= 3/2
Step-by-step explanation:
[tex] \frac{k}{4} = \frac{3}{8} \\ \\ k = \frac{12}{8} \\ k = \frac{3}{2} [/tex]