Best guess for the function is
[tex]\displaystyle f(x) = \sum_{n=1}^\infty \frac{x^n}{n^2}[/tex]
By the ratio test, the series converges for
[tex]\displaystyle \lim_{n\to\infty} \left|\frac{x^{n+1}}{(n+1)^2} \cdot \frac{n^2}{x^n}\right| = |x| \lim_{n\to\infty} \frac{n^2}{(n+1)^2} = |x| < 1[/tex]
When [tex]x=1[/tex], [tex]f(x)[/tex] is a convergent [tex]p[/tex]-series.
When [tex]x=-1[/tex], [tex]f(x)[/tex] is a convergent alternating series.
So, the interval of convergence for [tex]f(x)[/tex] is the closed interval [tex]\boxed{-1 \le x \le 1}[/tex].
The derivative of [tex]f[/tex] is the series
[tex]\displaystyle f'(x) = \sum_{n=1}^\infty \frac{nx^{n-1}}{n^2} = \frac1x \sum_{n=1}^\infty \frac{x^n}n[/tex]
which also converges for [tex]|x|<1[/tex] by the ratio test:
[tex]\displaystyle \lim_{n\to\infty} \left|\frac{x^{n+1}}{n+1} \cdot \frac n{x^n}\right| = |x| \lim_{n\to\infty} \frac{n}{n+1} = |x| < 1[/tex]
When [tex]x=1[/tex], [tex]f'(x)[/tex] becomes the divergent harmonic series.
When [tex]x=-1[/tex], [tex]f'(x)[/tex] is a convergent alternating series.
The interval of convergence for [tex]f'(x)[/tex] is then the closed-open interval [tex]\boxed{-1 \le x < 1}[/tex].
Differentiating [tex]f[/tex] once more gives the series
[tex]\displaystyle f''(x) = \sum_{n=1}^\infty \frac{n(n-1)x^{n-2}}{n^2} = \frac1{x^2} \sum_{n=1}^\infty \frac{(n-1)x^n}{n} = \frac1{x^2} \left(\sum_{n=1}^\infty x^n - \sum_{n=1}^\infty \frac{x^n}n\right)[/tex]
The first series is geometric and converges for [tex]|x|<1[/tex], endpoints not included.
The second series is [tex]f'(x)[/tex], which we know converges for [tex]-1\le x<1[/tex].
Putting these intervals together, we see that [tex]f''(x)[/tex] converges only on the open interval [tex]\boxed{-1 < x < 1}[/tex].
PLEASE HELP The linear functions f(x) and g(x) are represented on the graph, where g(x) is a transformation of f(x):
A graph with two linear functions; f of x passes through 1, 3 and 3, 13, and g of x passes through negative 1, 3 and 1, 13.
Part A: Describe two types of transformations that can be used to transform f(x) to g(x). (2 points)
Part B: Solve for k in each type of transformation. (4 points)
Part C: Write an equation for each type of transformation that can be used to transform f(x) to g(x). (4 points)
The transformation of f(x) to g(x) would be 6 units upwards
To transform f(x) to g(x), the equation is; g(x) = f(x) + 6
How to solve transformation problems?We are told that f of x passes through 1, 3 and 3, 13, and g of x passes through negative 1, 3 and 1, 13.
f(x) passes through (1,3) and (3, 13).
Thus;
slope is; m = (13 - 3) / (3 - 1) = 10/2 = 5
y-intercept is; b = y - mx
Thus;
b = 3 - 5*1 = -2
Equation of the line is;
f(x) = 5x - 2
g of x passes through negative 1, 3 and 1, 13. Thus, the slope is;
m = (13 - 3) / (1 + 1) = 10/2 = 5
y-intercept is; b = y - mx
Thus;
b = -1 - (5)(-1)
b = 4
Equation of the line is; g(x) = 5x + 4
part A: The transformation of f(x) to g(x) would be 6 units upwards
part B: k = 6
part C: To transform f(x) to g(x), the equation is;
g(x) = f(x) + k
g(x) = f(x) + 6
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Geometry: write formal proofs, ASAP!!!!
Answer:
By the definition of midpoints, AX and CX are congruent. By the definition of segment bisectors, X is the midpoint of BD, and therefore BX and DX are congruent. Since angle AXD and CXB are vertical angles, they are congruent by the vertical angles theorem. By SAS, triangles AXD and CXB are congruent. By CPCTC, angles A and C are congruent. By converse of alternate interior angles theorem, AD is parallel to CB.
Solid rock is an unlevered firm with an ebit of $10 million and an unlevered cost of capital of 12 percent. if the tax rate is 40 percent, what is the value of the firm?
The value of the firm is $50 million
How to determine the value of the firm?The given parameters are:
EBIT = $10 million
Tax Rate, Tc = 40%
Unlevered cost of capital, RU = 12%
The value of the firm is calculated as:
Value = EBIT*(1 - Tc)/Ru
This gives
Value = 10 million *(1 - 40%)/12%
Evaluate the expression
Value = 50 million
Hence, the value of the firm is $50 million
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Question
In the xy-plane, what is the y-intercept of the graph of the equation y = 6 (x − 1/2) (x+3)?
-9
-1/2
-1/2
3
9
Answer:
-9
Step-by-step explanation:
To find the y intercept, substitute 0 in for x.
So, the equation looks like this:
[tex]y=6(0-\frac{1}{2} )(0+3)[/tex]
Now, just solve for y and you will get y = -9
The y-intercept of the graph of the equation is y = -9 which is the correct answer that would be an option (A).
What is the equation of a line?The general equation of a line is y = mx + c
where m is the slope of the line and c is the intercept.
A linear equation is defined as an equation in which the highest power of the variable is always one.
Given equation as:
y = 6 (x - 1/2)(x+3)
To determine the y-intercept of the graph of the equation
Substitute the value x = 0 in the equation,
y = 6 (0 - 1/2)(0+3)
Apply the arithmetic operation and solve for y
y = -9
Hence, the y-intercept of the graph of the equation is y = -9.
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The purpose of statistical inference is to make estimates or draw conclusions about a.
When we make estimates of or draw conclusions about one or more characteristics of a population based upon the sample, we are using the process of statistical inference.
To estimate this sample to sample variance or uncertainty is the goal of statistical inference.What is the purpose of statistical inferences ?
To be able to make inferences about a population based on data from a sample is the goal of statistical inference. The process of statistical inference involves selecting a sample, gathering data from that sample, calculating a statistic from the data, and drawing conclusions about the population from that statistic.How is statistical inference used to draw conclusions?
Estimation and hypothesis testing are components of statistical inference (evaluating a notion about a population using a sample) (estimating the value or potential range of values of some characteristic of the population based on that of a sample).
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a 3 gallon jug of water cost $11.04. what’s the price per cup?
A man left rs 1,800,000 as inheritance.his heirs are 6 daughters and 2 sons.find the share of each heir
Answer:$225,000 each
Step-by-step explanation: Assuming that they each get an equal share of the inheritance, there are 8 shares of the inheritance so we divided the $1.8m by 8 which equals $225,000 each.
Highest Common Factor of 27 and 36.
Answer:
9
Step-by-step explanation:
listing the factors
27 : 1, 3, 9, 27
36 : 1, 2, 3, 4, 9, 12, 18, 36
the common factors are 1, 3, 9
the highest common factor is 9
Hi :)
The highest common factor is also known as the HCF
———————Let's find the HCF of [tex]\boldsymbol{27}[/tex] and [tex]\boldsymbol{36}[/tex].
[tex]\boldsymbol{Factors\:of\:27: 1, 3, 9, 27}[/tex]
[tex]\boldsymbol{Factors\:of\:36:1,2,3,4,6,9,12,18,36}[/tex]
HCF : [tex]\boldsymbol{9}[/tex]
[tex]\tt{Learn\:More;Work\;Harder}[/tex]
:)
WILL MARK U BRAINLIEST IF RIGHT PLS EXPLAIN
Answer:
166
Step-by-step explanation:
If the four lines are tangent to the circle, know that:
NA = KA
AD = LD
LC = MC
MB = NB
The question stated that:
NA = 18 --> KA is also 18
BN = 11 --> MB is also 11
AD = 34 --> LD is also 34
BC = 31 --> MC is 20 (Because BC is BM + MC and BM is 11)
We can just add these numbers.
(18+18) + (11 + 11) + (20+20) + (34+34)
= 36 + 22 + 40 + 68
= 76 + 90
= 166
The perimeter of ABCD is 166.
Answer:
130
Step-by-step explanation:
if 2 tangent segments are drawn to a circle from the same external point then they are congruent, then
AN = AK and KD = DL and MC = LC
then AN = AK = 18 and
KD = AD - AK = 34 - 18 = 16 , so KD = DL = 16
now BM = BN = 11 , then
AB = AN + BN = 18 + 11 = 29
MC = BC - BM = 31 - 11 = 20 and LC = MC = 20 , so
CD = LC + LD = 20 + 16 = 36
-----------------------------------------
Perimeter = AB + BC + CD + AD
= 29 + 31 + 36 + 34
= 130
When searching for the value 10 in the array [2, 3, 5, 6, 9, 13, 16, 19], a recursive binary search algorithm will return what?
When searching for the value 10 in the array [2, 3, 5, 6, 9, 13, 16, 19], a recursive binary search algorithm will return false since the element is not found in the array.
About Binary Search Algorithm
The binary search algorithm, applied on arrays are of recursive type. The broad strategy is to look at the middle item on the list. The procedure of the binary search algorithm is either terminated (key found), the left half of the list is searched recursively, or the right half of the list is searched recursively, depending on the value of the middle element.
The function carrying out the binary search algorithm in a code returns true if the desired element is found in the array, else returns false. Since the element 10 is not present in the given array: [2, 3, 5, 6, 9, 13, 16, 19], the binary search algorithm will return false.
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The data in the table represents the volume of helium, in cubic feet, inside a balloon relative to the elapsed time in minutes. What does the slope of the linear equation that models the data indicate?
The volume of helium inside the balloon decreased at a rate of 0.1 cubic feet per minute. The linear equation that represents the given data is y = -0.1x + 6 and its slope m = -0.1.
What is the equation of a line passing through two points?The equation of the line passing through the points (x1, y1) and (x2, y2) is
(y - y1) = m(x - x1).
Where m is the slope of the line and is calculated by
m = (y2 - y1)/(x2 - x1)
Calculation:The given data in the table represents the volume of helium (cu ft), inside a balloon relative to the elapsed time in minutes.
x: 5, 10, 15, 20, 25, 30, 35, 40
y: 5.5, 5, 4.5, 4, 3.5, 3, 2.5, 2
From this table, consider two points (5, 5.5) and (10, 5)
So, the equation of the line is
y- y1 = m(x - x1)
Where slope m = (y2 - y1)/(x2 - x1)
⇒ m = (5 - 5.5)/(10 - 5)
⇒ m = -0.5/5
∴ m = -0.1
Thus, the equation is
y - 5.5 = -0.1(x - 5)
⇒ y - 5.5 = -0.1x + 0.5
⇒ y = -0.1x + 0.5 + 5.5
⇒ y = -0.1x + 6.0
∴ y = -0.1x + 6
Since the slope of the line is negative, the volume of the helium inside the balloon decreased at a rate of 0.1 cubic ft per minute.
So, option 2 is correct.
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Find the general solution of the given higher-order differential equation. y''' − 8y'' − 9y' = 0
The general solution of the given higher-order differential equation y''' − 8y'' − 9y' = 0 is , [tex]y = C_{1} + C_{2} e^{8x} + C_{3}e^{-x}[/tex]
Given higher-order differential equation = y''' − 8y'' − 9y' = 0
We have to find the general solution of the above differential equation
The auxiliary equation for the above differential equation will be : [tex]m^{3} - 8m^{2} - 9m = 0[/tex]
Solve the equation :- [tex]m^{3} - 8m^{2} -9m = 0[/tex]
[tex]m ( m^{2} - 8m - 9 ) = 0[/tex]
m ( m - 8 ) ( m + 1 ) = 0
m = 0 , m = 8 , m = -1
Then the general solution will be like :
[tex]y = C_{1} e^{0x} + C_{2} e^{x} +C_{3}e^{-1x}[/tex]
The above solution can be written as:
[tex]y = C_{1} +C_{2} e^{8x} +C_{3} e^{-x}[/tex]
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Multiply (x - 3)(x2 + 7x - 2)
Answer:
x^3+4x^2−23x+6
Step by Step:
(x−3)(x2+7x−2)
=(x+−3)(x2+7x+−2)
=(x)(x2)+(x)(7x)+(x)(−2)+(−3)(x2)+(−3)(7x)+(−3)(−2)
=x3+7x2−2x−3x2−21x+6
=x3+4x2−23x+6
How many different 2-digit numbers are there with the following property:the tenth digit is greater than the units digit?
Answer:
45
Step-by-step explanation:
Simply start by listing out the numbers from smallest to largest.
We start out with the smallest number following this information being the number 10. The next are 21 and 20. The next are 32, 31, and 30. We can start to see a pattern. If the ten's digit is n, then there are n numbers within that ten-group that have a greater 10's digit than unit's digit. The largest ten-group (90 to 98) has 9 integers following the rules established. Thus, the number of 2-digit integers that have ten's digits greater than units digit is 1+2+3+...+9=45
Geometry: complete this proof of theorem 18, ASAP!!!
(Theorem 18: if the side of one angle are parallel to the sides of another angle, right side to left side and left side to right side, then the angle are supplementary)
Supplementary angles are angles that can be summed up to give [tex]180^{o}[/tex], which is the value of the sum of angles on a line.
The required statements and reasons are given below:
Two or more lines are said to be parallel if there is no measure of the angle between them or the measure of the angle between them is [tex]180^{o}[/tex]. Thus the lines would never meet at any point even when extended to infinity.
While supplementary angles are angles that can be summed up to give [tex]180^{o}[/tex], which is the value of the sum of angles on a line.
STATEMENTS REASONS
1. < A and <B with AD ║BE, AC ║BF Given.
2. AD, AC, BE, BF Straight line.
3. <A ≅ <1 Congruent property of
vertically opposite angles.
4. AD + AC ≅ DC Addition property of parts
BE + BF ≅ EF of a given line segment.
5. <A + <B ≅ [tex]180^{o}[/tex] Addition property of
supplementary angles.
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there are 6 girls and 8 boys in Mr. Stevenson's class the ratio of girls to boys in the class can be expressed as the simplified ratio ?/4 what is the missing number ?
Answer: 3
Step-by-step explanation:
The missing number in the simplified ratio ?/4 is 3.
Here, we have to find the missing number in the simplified ratio ?/4, we need to determine the ratio of girls to boys in Mr. Stevenson's class.
Given that there are 6 girls and 8 boys in the class, the ratio of girls to boys can be expressed as:
Number of girls : Number of boys = 6 : 8
To simplify the ratio, we can divide both the number of girls and the number of boys by their greatest common divisor (GCD).
In this case, the GCD of 6 and 8 is 2.
Dividing both 6 and 8 by 2, we get:
Number of girls : Number of boys = 6/2 : 8/2 = 3 : 4
So, the simplified ratio of girls to boys is 3/4.
Therefore, the missing number in the simplified ratio ?/4 is 3.
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Select the graph of the solution. Click until the correct graph appears.
| x | + 1 > 2
Answer: Third Graph
Step-by-step explanation:
Given inequality
|x| + 1 > 2
Subtract 1 on both sides
|x| + 1 - 1 > 2 - 1
|x| > 1
Simplify the absolute value sign
x > 1 or x < -1
Apply the inequality to the graph
Since it is less than or greater than, the circle should be hollow
Since it is less than -1 or greater than 1, one line should point to the left and another should point to the right
Therefore, it should be the Third Graph
(9⋅10
9
)⋅(−2⋅10
−3
)=left parenthesis, 9, dot, 10, start superscript, 9, end superscript, right parenthesis, dot, left parenthesis, minus, 2, dot, 10, start superscript, minus, 3, end superscript, right parenthesis, equals
Choose 1 answer:
Choose 1 answer:
(Choice A, Checked)
A
-18\cdot 10^6−18⋅10
6
minus, 18, dot, 10, start superscript, 6, end superscript
(Choice B)
B
-18 \cdot 10^{5}−18⋅10
5
minus, 18, dot, 10, start superscript, 5, end superscript
(Choice C)
C
18\cdot 10^{-6}18⋅10
−6
18, dot, 10, start superscript, minus, 6, end superscript
(Choice D)
D
18\cdot 10^{-5}18⋅10
−5
Answer:
A. -18·10^6
Step-by-step explanation:
The rules of exponents apply to powers of 10 used in scientific notation. The associative and commutative properties of multiplication also apply.
Application(a^b)(a^c) = a^(b+c)
The numerical product is ...
[tex](9\cdot10^9)\cdot(-2\cdot10^{-3})=(9)(-2)(10^{9-3})=\boxed{-18\cdot10^6}[/tex]
__
Additional comment
Expressed in scientific notation, the result would be ...
[tex]-1.8\cdot10^7[/tex]
Your calculator can perform this multiplication for you.
find the surface area of composite figure. use 3.14 for π. round to the nearest tenth
Answer:
628.0 mm²
Step-by-step explanation:
The total surface area of the figure is the sum of the inside lateral area, the outside lateral area, and the area of the donut bases.
Lateral areaThe lateral area of a cylinder is ...
A = 2πrh
The total lateral area of the inside and outside cylinders is ...
A = 2π(r1)h +2π(r2)h = 2π(r1 +r2)h
A = 2(3.14)(3 mm +7 mm)(6 mm) = 376.8 mm²
Base areaThe area of one donut base is the product of the centerline length and the width.
A = πdw = (3.14)(7 mm +3 mm)(4 mm) = 125.6 mm²
Total areaThe total surface area of the composite figure is the sum of its lateral area and the area of the two bases.
surface area = 376.8 mm² +2×125.6 mm² = 628.0 mm²
__
Additional comment
The radius of the centerline of the base donut is the average of the inside and outside radii: half their sum. The diameter of the centerline circle is twice that average radius, so is equal to the sum of the inside and outside radii. This is the value we used above.
The width of the donut is the difference in the radii.
The product of the sum and difference is the same as the difference of the squares of the radii. That difference of squares would be what you have if you compute the overall area and subtract the inner area.
Zoey read a novel in 30 days she made it a habit to read 13 pages everyday for the first 20 days. if in the last 10 days she reads 6 pages everyday how many pages are in the book?
Answer:
320 pages.
Step-by-step explanation:
We can simply multiply and add to find how many pages are in the book.
So we have (13 pg * 20 d) + (6 pg * 10 d) = 260 + 60 = 320.
Another way of solving is to set up a proportion for both the 20 and 10 days, find how many pages are read on these days, and add:
[tex]\frac{13pg}{1d}=\frac{xpg}{20d}\\ x=260pages\\\\\frac{6pg}{1d} =\frac{xpg}{10d}\\ x=60pg\\260+60=320[/tex]
Consider the three arithmetic sequences. sequence i: 9, 13, 17, 21, . . . sequence ii: 117, 120, 123, 126, . . . sequence iii: 54, 61, 68, 75, . . . which lists the sequences in order from least common difference to greatest common difference?
Answer: sequence II, sequence I, sequence III
A clock is showing the correct time at 8 am. if it gains four minutes in every hour what time will it be showing five and a half hours later ?
Answer:
1:52 PM
Step-by-step explanation:
It gains 4 min every hour so it gains 4 * 5.5 = 22 minutes
8 oclock + 5.5 hrs + 22 minutes =
1:52
which expression is equivalent to (60x^(20)y^(24))/(30x^(10)y^(12)
a. 2x^(2)y^(2)
b. 2x^(10)y^(12)
c. 30x^(2)y^(2)
d.30x^(10)y^(12)
Answer:
b. 2x^10y^12.
Step-by-step explanation:
(60x^(20)y^(24))/(30x^(10)y^(12)
60 / 30 = 2
x^20 / x^10 = x^(20-10) = x^10
y^(24) / y^(12) = y^(24-12) = y^12.
Thus, the answer is:
2x^10y^12.
Answer:
b. 2x^(10)y^(12)
Step-by-step explanation:
(60x^(20)y^(24))/(30x^(10)y^(12)
when there is division we simplify that by subtracting the power
by using rules of indices
[tex] \frac{x^a}{x^b}=x^{a-b}[/tex]
and number can be divided easily
[tex] \frac{60}{30}*\frac{x^{20}}{x^{10}}*\frac{y^{24}}{y^{12}}[/tex]
[tex] 2*x^{20-10}y^{24-12}[/tex]
[tex] 2x^{10}y^{12}[/tex]
so answer is b. 2x^(10)y^(12)
Determine the unknown length or angle measurement. Round each answer to the nearest whole number.
Answer:
a) 52
b)115
Step-by-step explanation:
hich value, when placed in the box, would result in a system of equations with infinitely many solutions?
y = -2x + 4
6x + 3y =
The value placed in the box that makes the system of equation with infinitely many solution is 12.
How to solve an infinitely many solution equation?An infinite solution has both sides equal. For example, 6x + 2y - 8 = 12x +4y - 16. If we simplify the equation we will notice both sides are equal. This means the equation has an infinitely many solution.
Hence,
y = -2x + 4
Therefore,
6x + 3y = 12
divide the equation(ii) by 3
2x + y = 4
y = -2x + 4
Therefore, both equation are equal if the the box is filled with 12. This means for the value placed in the box that makes the system of equation with infinitely many solution is 12.
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which relation is a function?
The relation that is a function is:
b. y = 2x² - 3x + 7
When a relation is a function?A relation is a function if each value of the input is mapped to only one value of the output.
Hence, when both x and y are squared, we have that [tex]x = \pm ay[/tex], hence it is not a function. This is the case for items a and c.
For item d, we have that the relation can be simplified as follows:
x = -y² + 3y
x = y(-y + 3)
The solution above, is associated to two values of y, hence it is also not a function. Then the function is given by:
b. y = 2x² - 3x + 7
In the solution, it can be seen that for each input x, only one value of y can be generated.
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Which power of Congress is the most important?
Select one:
O a. the authority to regulate commerce
O b. the authority to establish the federal courts.
O c. the authority to make laws
O d. the authority to coin money
Round each fraction to help you estimate the solution for the following equation:
nine tenths minus eight tenths equals
0
one half
1
one and one half
Answer:
Step-by-step explanation:
answer is 1/2
Which equation below
represents exponential
growth?
a. y=3x^2-5
b. y = 3(2/5)^x
c. y=3(5)^x
d. y=3x-5
There is a balcony that forms part of a circle around a stage, and they need to put up a safety railing. How long of a railing do they need if the radius of the circle is 40 feet, and the arc takes up 45°? Use 3.14 for pi.
The length of the railing needed is 31.4 feet
Calculating the length of an arcFrom the question, we are to determine the length of railing needed
To determine the length of railing needed, we will determine the length of the arc formed by the balcony
Using the formula,
l = θ/360° × 2πr
Where l is the length of the arc
θ is the angle subtended by the arc
and r is the radius
From the given information,
θ = 45°
r = 40 feet
Putting the parameters into the equation, we get
l = 45°/360° × 2 × 3.14 × 40
l = 1/8 × 2 × 3.14 × 40
l = 2 × 3.14 × 5
l = 31.4 feet
Hence, the length of the railing needed is 31.4 feet
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