To prove that the given series (sin x)2 x 2n +1 n=1 is uniformly convergent in R, we'll make use of the M-test for uniform convergence.
Let f(x,n) = (sin x)2 x 2n +1 n=1
Let Mn = sup[ f(x,n) ] ( x in R )
To find Mn, we differentiate f(x,n) w.r.t. x, set it to zero to get its critical points and find its maximum at these critical points.
Now, f'(x,n) = 2 sin(x) cos(x) x 2n+1 - (sin(x))2 (2n + 1)x2n = sin(x)x2n[ 2 cos(x) - (2n+1)(sin(x)/x) ]= 0 either when sin(x) = 0, i.e. at x = nπ for any integer n, or when cos(x) = (2n+1)(sin(x)/x) / 2 (for non-zero values of x).
The values of x for which cos(x) = (2n+1)(sin(x)/x) / 2 are the critical points.
But since | (2n+1)(sin(x)/x) / 2 | <= | 2n + 1 | / 2, we have | cos(x) | <= | 2n + 1 | / 2.
Hence, the critical values of cos(x) for any given value of n are between - (2n+1)/2 and (2n+1)/2 (note that these values depend on n).
Hence, we have:
Mn = sup[ f(x,n) ] ( x in R )<= sup[ f(x,n) ] ( x in [- (2n+1)/2, (2n+1)/2] )<= f((2n+1)/2,n) = [(2n+1)/2]2n+1 sin((2n+1)/2)
This last step uses the fact that sin(x) <= 1 for all x and (sin(x))2 <= 1 for all x, as well as the observation that f(x,n) is odd, so its maximum will occur at x = (2n+1)/2.
Hence, we have:
Mn <= [(2n+1)/2]2n+1 sin((2n+1)/2)
Now, we claim that this upper bound of Mn goes to zero as n goes to infinity.
To show this, we use the fact that sin(x)/x -> 1 as x -> 0. Hence, sin((2n+1)/2) / (2n+1)/2 -> 1 as n -> infinity. Thus, [(2n+1)/2]2n+1 sin((2n+1)/2) -> 0 as n -> infinity.
Therefore, by the M-test, the series (sin x)2 x 2n +1 n=1 is uniformly convergent in R.
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1. (1 point) Let x be a real number. Show that a (1 + x)2n > 1+ 2nx for every positive integer n.
For a real number x, by using mathematical induction it is shown that a[tex](1 + x)^{2n}[/tex] > 1 + 2nx for every positive integer n.
To prove the inequality a[tex](1 + x)^{2n}[/tex] > 1 + 2nx for every positive integer n, we will use mathematical induction.
The inequality holds true for n = 1, and we will assume it is true for some positive integer k.
We will then show that it holds for k + 1, which will complete the proof.
For n = 1, the inequality becomes a[tex](1 + x)^2[/tex] > 1 + 2x.
This can be expanded as a(1 + 2x + [tex]x^2[/tex]) > 1 + 2x, which simplifies to a + 2ax + a[tex]x^2[/tex] > 1 + 2x.
Now, let's assume the inequality holds true for some positive integer k, i.e., a[tex](1 + x)^{2k}[/tex] > 1 + 2kx.
We need to prove that it holds for k + 1, i.e., a[tex](1 + x)^{2(k+1)}[/tex] > 1 + 2(k+1)x.
Using the assumption, we have a[tex](1 + x)^{2k}[/tex] > 1 + 2kx.
Multiplying both sides by [tex](1 + x)^2[/tex], we get a[tex](1 + x)^{2k+2}[/tex] > (1 + 2kx)[tex](1 + x)^2[/tex].
Expanding the right side, we have a[tex](1 + x)^{2k+2}[/tex] > 1 + 2kx + 2x + 2k[tex]x^2[/tex] + 2[tex]x^2[/tex].
Simplifying further, we get a[tex](1 + x)^{2k+2}[/tex] > 1 + 2(k+1)x + 2k[tex]x^2[/tex] + 2[tex]x^2[/tex].
Since k and x are positive, 2k[tex]x^2[/tex] and 2[tex]x^2[/tex] are positive as well.
Therefore, we can write a[tex](1 + x)^{2k+2}[/tex] > 1 + 2(k+1)x + 2k[tex]x^2[/tex] + 2[tex]x^2[/tex] > 1 + 2(k+1)x.
This proves that if the inequality holds for some positive integer k, it also holds for k + 1.
Since it holds for n = 1, it holds for all positive integers n by mathematical induction.
Therefore, we have shown that a[tex](1 + x)^{2n}[/tex] > 1 + 2nx for every positive integer n.
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Having an error of 0.01, a confidence level of 95% with a value p = 0.37 Determine: a) Z-value b) Sample size
a) The Z-value corresponding to a confidence level of 95% can be determined as 1.96.
b) To determine the required sample size, we need additional information such as the population size or an estimated proportion. Without this information, we cannot calculate the sample size.
a) The Z-value represents the number of standard deviations a data point is from the mean in a standard normal distribution. For a confidence level of 95%, which corresponds to a 5% significance level, the critical Z-value is approximately 1.96. This value can be obtained from a Z-table or using statistical software.
b) To calculate the required sample size, additional information is needed, such as the population size or an estimated proportion. The sample size formula takes into account factors such as the desired margin of error, confidence level, and variability. Without these details, it is not possible to determine the sample size accurately.
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The points where a graph crosses the x- and y-axis are called the __
Answer:
x-intercept
y-intercept
Step-by-step explanation:
Answer:
x-intercept y-intercept
PLSSS HELP IMMEDIATELY!!!! i’ll mark brainiest if u don’t leave a link!
Answer:
grab objects
Step-by-step explanation:
barrels and things like that would most likely grab objects so the fish can have room to swim and stuff
The important difference to note for the scales of measurement and how they are analyzed is whether they involve Oratios, intervals categories, ration O numbers, categories O numbers, intervals as responses on the scale.
The important difference to note for the scales of measurement and how they are analyzed is whether they involve ratios, intervals, categories or numbers. The scales of measurement can be divided into four types: nominal, ordinal, interval, and ratio.
Nominal scales use categories or numbers to group data, but these categories or numbers have no inherent order or value. Examples of nominal scales include gender, race, or eye color. Ordinal scales use categories or numbers to group data, but these categories or numbers have a specific order or rank. Examples of ordinal scales include educational attainment, income, or level of agreement on a survey question.
Interval scales use numbers as responses on the scale, but the distance between the numbers is not meaningful. Examples of interval scales include temperature measured in Celsius or Fahrenheit, or IQ scores. Ratio scales use numbers as responses on the scale, but the distance between the numbers is meaningful and there is a true zero point. Examples of ratio scales include height, weight, or income.
In summary, the important difference to note for the scales of measurement and how they are analyzed is whether they involve categories or numbers, and whether the numbers have a specific order or rank, a meaningful distance between them, or a true zero point.
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question in screenshot
Answer:
10
Step-by-step explanation:
use pythagorean theorm
√(8^2+6^2) = 10
What is 0.0003246 expressed in scientific notation?
A.
32.46
×
10
−
5
B.
3.246
×
10
−
4
C.
3.246
×
10
4
D.
32.46
×
10
5
Answer:
B
Step-by-step explanation:
the notation answer would also be 3.246 × 10-4 i believe :)
PLEASE HELP ME OUT! QUICK POINTS FOR YOU!
All information needed can be found in the image below
Thank you in advance.
Answer:
5π
Step-by-step explanation:
just need to find half of the circumference
Please answer correctly!
Answer:
B. 1296 m^3
Step-by-step explanation:
To find the volume of a square pyramid, multiply the base by itself twice, then divide the height by 3. After getting both answers, multiply the answers.
In this case, the base is 18.
The height is 12.
First, we must multiply the base by itself twice.
18⋅18 = 324.
Next, divide the height by 3.
12/3 = 4.
Now that we have both answers, we multiply them.
324 ⋅ 4 = 1,296.
Therefore, 1,296 cm^3 is the volume of the square pyramid.
$297.89 with a 9.5% tax
What should you do first when you simplify the expression below? (5+4)x3
Answer:
5 + 4 snice it's inside the circle and when you get your answer you times it with 3
Step-by-step explanation:
( 5 + 4 ) x 3
9 x 3
27
(5+4) x 3 = 27
Select all the properties of a right triangle.
A. Has a Right Angle
B. Has 1 Obtuse Angle
C. Has Parallel Lines
D. Has Perpendicular Lines
E. Has 3 Acute Angles
If u are an Expert PLzzz Answer Thisss!
Answer:
Sorry sis don't know the meaning
F(x_1,x_2,x_3) = (y_1,y_2,y_3) Set
(1) x_1/ (x_1 + x_2 + x_3) = y_1
(2) x_2/ (x_1 + x_2 + x_3) = y_2
(3) x_3/ (x_1 + x_2 + x_3) = y_3
1. Prove that F is injective.
2. Without appealing to the Inverse Function Theorem, find the investment directly.
3. Find the domain of F
4. Find the range of F, which is the domain of F^-1.
5. Explain why J_f(x) 6=0 when x € Dom (F)
1. Proof that F is injective. Suppose that two different elements in the domain of F have the same image; that is, if x and y are elements of the domain of F and F(x)= F(y). We need to show that x=y. Let F(x) = F(y). This means that y1= x1/ (x1 + x2 + x3) = y1, y2= x2/ (x1 + x2 + x3) = y2 and y3= x3/ (x1 + x2 + x3) = y3.Now adding (1), (2), and (3), we have:y1+y2+y3= x1/ (x1 + x2 + x3) + x2/ (x1 + x2 + x3) + x3/ (x1 + x2 + x3)But this is equal to 1, therefore,x1 + x2 + x3= y1 + y2 + y3 = 1, or, equivalently, y1= 1- y2 - y3x1= (1- y2 - y3)(x1 + x2 + x3) = (1-y2 - y3), x2= y2(x1 + x2 + x3) = y2, and x3= y3(x1 + x2 + x3) = y3Thus, we have constructed an element of the domain of F, with different elements of the domain, that have the same image. Therefore, F is injective.
2. Find the investment directly without appealing to the Inverse Function Theorem. F(x1,x2,x3) = (y1,y2,y3)So, x1= (1- y2 - y3), x2= y2, and x3= y3Thus, the inverse of F is F-1(y1,y2,y3)= ((1-y2-y3),y2,y3)3. The domain of F. The domain of F is the set of all three-tuples, F(x1,x2,x3) where 0 ≤ xi ≤ ∞, and where at least one xi is positive. That is, Dom (F)={(x1,x2,x3)|x1≥0, x2≥0, x3≥0 and (x1,x2,x3)≠(0,0,0)}4. The range of F, which is the domain of F-1. The range of F is the set of all three-tuples, F(y1,y2,y3) where 0 ≤ yi ≤ 1, and where at least one yi is positive.
That is, Rng(F)={(y1,y2,y3)|y1≥0, y2≥0, y3≥0 and y1+y2+y3=1}5. We have J_f(x) = ∣∣ ∂(y1,y2,y3) /∂(x1,x2,x3) ∣∣= ∣∣ ∂y1/∂x1 ∂y1/∂x2 ∂y1/∂x3 ∂y2/∂x1 ∂y2/∂x2 ∂y2/∂x3 ∂y3/∂x1 ∂y3/∂x2 ∂y3/∂x3 ∣∣= ∣∣ (1 / (x1 + x2 + x3) ) - (x1/ (x1 + x2 + x3)2) - (x1/ (x1 + x2 + x3)2) 0 1 / (x1 + x2 + x3) - (x2/ (x1 + x2 + x3)2) 0 0 1 / (x1 + x2 + x3) - (x3/ (x1 + x2 + x3)2) ∣∣= (1 / (x1 + x2 + x3) )((1 - y2 - y3) (1 - y2 - y3 - y3) - y2 (1 - y2 - y3) - y3(1 - y2 - y3)) = (1 / (x1 + x2 + x3) )((1 - y2 - y3 - y2 + 2y2y3 + y2 - y3 - 2y2y3 + y3 - y2y3 - y3 + y2y3)) = 1 / (x1 + x2 + x3) which is non-zero in the domain of F. Therefore, J_f(x) ≠ 0 when x ∈ Dom (F).
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I ask for your help fellow strugglers
Answer: C is false
Step-by-step explanation: If B is correct and C is saying otherwise that must mean it's the only false choice :) brainliest would be appreciated :)
8^15÷8^−3 it needs to like this 43^5 I can't find out the answer
Answer:
Hello!
The answer is 8^18.
Step-by-step explanation:
Answer:
8^12
Step-by-step explanation:
Exponents are confusing at first. You subtract them when dividing, and add them when multiplying, if the base number is the same.
Picture 8 to the 15th as a numerator of:
8x8x8x8x8x8x8x8x8x8x8x8x8x8x8 over a denominator of
8x8x8
To solve that, you'd cancel out 3 of the top 8s and the three bottom 8s, marked in bold to be easier to see. This leaves you with
8^12
Does that help?
Happy to answer questions.
Find the lateral area of this square
based pyramid.
10
ft
10 ft
[ ? jft?
Answer:
200
Step-by-step explanation:
happy to help
The lateral area of square based pyramid is 200 square feet
What is Three dimensional shape?a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.
The given square base pyramid has four faces of triangles and one square shape base
The lateral surface area of pyramid formula 4 × (1/2)bl,
b = side length of the base, and
l = slant height
Lateral surface area =2 bl
=2×10×10
=200 square feet
Hence, the lateral area of square based pyramid is 200 square feet
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Please answer these 3 answers correctly
Answer: 3. Tan; 4. 4.76; 5. 12
Step-by-step explanation: First question: SOH CAH TOA; if you draw a picture with the given information you know the Opposite and Adjacent sides to the upper left angle, or OA which is also Tan according to Soh Cah Toa. Second question: Use inverse tan(1/12) to solve. Third question: Same idea as the last question but use inverse sin(1/5)
I need help with this please help me.
Answer:
a) $93, $117.80, $86.80, $68.20
b) $365.80
Step-by-step explanation:
a)
Add 12 hours to each p.m. time to make the subtraction of hours easier.
Also, use decimal numbers to do the subtraction. Remember that 0:30 means 30 minutes which is 0.5 hour, so 7:30 is the same as 7.5 as a decimal.
Monday: 8:00 to 15:30
15:30 - 8:00 = 1.5 - 8 = 7.5 hours
7.5 hours * $12.40/h = $93
Wednesday: 7:30 to 17:00
17:00 - 7:30 = 17 - 7.5 = 9.5 hours
9.5 hours * $12.40/h = $117.80
Friday: 9:00 to 16:00
16:00 - 9:00 = 16 - 9 = 7 hours
7 hours * $12.40/h = $86.80
Saturday: 10:30 to 16:00
16:00 - 10:30 = 16 - 10.5 = 5.5 hours
5.5 hours * $12.40/h = $68.20
b) Add all the daily amounts above.
$93 + $117.80 + $86.80 + $68.20 = $365.80
Always in the game and never played by the rules (rules)
Tried to let me leave (leave), fell down to my knees (knees)
Picked myself up and turned my back to the breeze, oh-oh
Spent a milli' on a milli', mama look at me (oh)
With all these diamond chokers, man, it's gettin' hard to breathe
If you made this Very good job. I like it, and its hard to make a song i like. i like about 60 songs total out of the 10000+ ik of
please help due in a few hours!
Answer:
54 cm²
Step-by-step explanation:
1 cm= 3 m
2 cm= 6 m
3 cm= 9 m
9×6=54
Area of bedroom= 54 cm²
what is the area of 1 1/5 width and 1 1/3 length
Answer:
1.6 unit^2 (dec. form) or 1 3/5 unit^2 (frac. form)
Step-by-step explanation:
(1 1/5)(1 1/3)
(6/5)(4/3)
1.6 unit^2 (dec. form) or 1 3/5 unit^2 (frac. form)
.222222222 as a fraction Please help
Answer:
222222222/1000000000
Step-by-step explanation:
A game has a 10-sided die. What is the probability of rolling a number less than 3 or an odd number? All answers should be in FRACTION form ONLY.
The probability of rolling a number less than 3 or an odd number is 3/5 in fraction form.
To compute the probability of rolling a number less than 3 or an odd number, we need to calculate the probability of each event separately and then subtract the probability of their intersection.
The probability of rolling a number less than 3 is 2/10, as there are two numbers (1 and 2) that satisfy this condition out of the ten possible outcomes.
The probability of rolling an odd number is 5/10, as there are five odd numbers (1, 3, 5, 7, and 9) out of the ten possible outcomes.
To compute the probability of their intersection (rolling a number less than 3 and an odd number), we observe that there is only one number (1) that satisfies both conditions.
Therefore, the probability of their intersection is 1/10.
To compute the probability of rolling a number less than 3 or an odd number, we sum the probabilities of each event and subtract the probability of their intersection:
Probability of rolling a number less than 3 or an odd number = (2/10) + (5/10) - (1/10) = 6/10 = 3/5.
Therefore, the probability of rolling a number less than 3 or an odd number is 3/5.
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Let C41 be the graph with vertices {0, 1,..., 40} and edges (0-1), (1-2),..., (3910), (100), and let K41 be the complete graph on the same set of 41 vertices. You may answer the following questions with formulas involving exponents, binomial coefficients, and factorials. (a) How many edges are there in K41? (b) How many isomorphisms are there from K41 to K41? (c) How many isomorphisms are there from C41 to C41? (d) What is the chromatic number (K41)? (e) What is the chromatic number (C41)? (f) How many edges are there in a spanning tree of K41? (g) A graph is created by adding a single edge between nonadjacent vertices of a tree with 41 vertices. What is the largest number of cycles the graph might have? (h) What is the smallest number of leaves possible in a spanning tree of K41? i) What is the largest number of leaves possible in a in a spanning tree of K41? G) How many spanning trees does C41 have?
(a) The complete graph K41 has 820 edges. This can be calculated using the formula for the number of edges in a complete graph, which is given by the expression (n(n-1))/2, where n is the number of vertices. Substituting n = 41, we get (41(41-1))/2 = 820.
(b) The number of isomorphisms from K41 to itself is equal to the number of permutations of the vertices. This can be calculated as 41!, which represents the number of ways to arrange the vertices of K41.
(c) The graph C41 is not isomorphic to itself because it has a specific edge pattern. Thus, there are no isomorphisms from C41 to itself.
(d) The chromatic number of K41 is equal to its number of vertices, which is 41. This is because each vertex can be assigned a unique color, and no two adjacent vertices share the same color in a complete graph.
(e) The chromatic number of C41 is 2. This is because C41 contains a Hamiltonian cycle, which is a cycle that visits each vertex exactly once. A Hamiltonian cycle can be colored with only two colors, where adjacent vertices are assigned different colors.
(f) A spanning tree of K41 is a connected acyclic subgraph that includes all the vertices of K41. The number of edges in a spanning tree of K41 is equal to the number of vertices minus 1, which is 41 - 1 = 40.
(g) If a single edge is added between nonadjacent vertices of a tree with 41 vertices, the largest number of cycles the graph might have is 41. This can occur when the new edge connects two vertices that are at maximum distance from each other in the original tree, resulting in a new cycle.
(h) The smallest number of leaves possible in a spanning tree of K41 is 1. This can be achieved by removing all but one edge from K41, resulting in a single leaf node.
(i) The largest number of leaves possible in a spanning tree of K41 is 40. This can be achieved by removing all but one vertex from K41, resulting in a single vertex connected to 40 leaf nodes.
(g) The number of spanning trees that C41 has can be calculated using Cayley's formula, which states that a complete graph with n vertices has nn-2 spanning trees. Substituting n = 41, we get 4141-2 = 240 spanning trees for C41.
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need help with this one lol
Answer: 5 + 47 = 108
Step-by-step explanation: dont worry it works
Answer:
pythagoras thoerem
Step-by-step explanation:
y squared = 10 squared - 7 squared
y = 100 - 49= 51
y square root of 51 =7.1
simplify log(16x²) ± 2㏒(1÷×)
Answer: just simplify condense it then your answer will be log(16)
A stem and leaf plot titled High Temperatures in degrees Fahrenheit. The stems are 4, 5, 6, 7. The first column of leaves are 9, 2, 0, 2. The second column of leaves are blank, 4, 1, 3. The third column of leaves are blank, 4, blank, blank. The fourth column of leaves are blank, 6, blank, blank. The fifth column of leaves are blank, 8, blank, blank. The stem and leaf plot shows high temperatures recorded each day at the beginning of the month. Which temperature occurs the most frequently
Answer:
The temperature with the highest frequency is 54
Step-by-step explanation:
Given
The above data
Required
The temperature with the highest frequency
The first step, is to plot the stem and leaf plot.
[tex]\begin{array}{cc}{4} & {9\ \ } & {5} & {2\ 4\ 4\ 6\ 8} & {6} & {0\ 1\ } & {7} & {2\ 3\ } \ \end{array}[/tex]
Next, is to identify the leaf with the highest frequency.
The leaf is 4 (with frequency 2)
Next, is to identify the accompanying stem of the leaf
The stem is 5
Hence, the temperature with the highest frequency is 54
Answer:
B
Step-by-step explanation:
edg 2021
Please help no links
Answer: Big rectangle shades 1/4+1/2
Step-by-step explanation:
So have a Big
The diameter of a circle is 6 kilometers. What is the area?
d=6 km
Give the exact answer in simplest form.
square kilometers
Answer:
28.27 rounded orrrr 28.27433388 not rounded.
Step-by-step explanation:
area of circle=πr^2
radius=3 km
3^2=9
9*π
Step-by-step explanation:
Area of a circle = πr²
Radius (r) =1/2 Diameter
=60/2
=30km
Area = π x (30)²
= π x 900
= 2027km²
Can someone please help me please I really need help please answer it correctly
Answer:
Princeton FloristLet the total charge is y, the number of small arrangements is x.
Total charge will be:
y = 13x + 47Chad's FlowersTotal charge will be:
y = 17x + 35Since the total charge is same in both shops, we have:
13x + 47 = 17x + 35Solve for x:
17x - 13x = 47 - 354x = 12x = 3Total cost is:
13*3 + 47 = 39 + 47 = 86Small arrangements = 3, cost = $86