[tex]\large \rm \sum \limits_{n = 0}^ \infty \frac{( { - 1)}^{1 + 2 + 3 + \dots + n} }{(2n + 1 {)}^{2} }[/tex]
Answers
The sum we want is
[tex]\displaystyle \sum_{n=0}^\infty \frac{(-1)^{T_n}}{(2n+1)^2} = 1 - \frac1{3^2} - \frac1{5^2} + \frac1{7^2} + \cdots[/tex]
where [tex]T_n=\frac{n(n+1)}2[/tex] is the n-th triangular number, with a repeating sign pattern (+, -, -, +). We can rewrite this sum as
[tex]\displaystyle \sum_{k=0}^\infty \left(\frac1{(8k+1)^2} - \frac1{(8k+3)^2} - \frac1{(8k+7)^2} + \frac1{(8k+7)^2}\right)[/tex]
For convenience, I'll use the abbreviations
[tex]S_m = \displaystyle \sum_{k=0}^\infty \frac1{(8k+m)^2}[/tex]
[tex]{S_m}' = \displaystyle \sum_{k=0}^\infty \frac{(-1)^k}{(8k+m)^2}[/tex]
for m ∈ {1, 2, 3, …, 7}, as well as the well-known series
[tex]\displaystyle \sum_{k=1}^\infty \frac{(-1)^k}{k^2} = -\frac{\pi^2}{12}[/tex]
We want to find [tex]S_1-S_3-S_5+S_7[/tex].
Consider the periodic function [tex]f(x) = \left(x-\frac12\right)^2[/tex] on the interval [0, 1], which has the Fourier expansion
[tex]f(x) = \frac1{12} + \frac1{\pi^2} \sum_{n=1}^\infty \frac{\cos(2\pi nx)}{n^2}[/tex]
That is, since f(x) is even,
[tex]f(x) = a_0 + \displaystyle \sum_{n=1}^\infty a_n \cos(2\pi nx)[/tex]
where
[tex]a_0 = \displaystyle \int_0^1 f(x) \, dx = \frac1{12}[/tex]
[tex]a_n = \displaystyle 2 \int_0^1 f(x) \cos(2\pi nx) \, dx = \frac1{n^2\pi^2}[/tex]
(See attached for a plot of f(x) along with its Fourier expansion up to order n = 10.)
Expand the Fourier series to get sums resembling the [tex]S'[/tex]-s :
[tex]\displaystyle f(x) = \frac1{12} + \frac1{\pi^2} \left(\sum_{k=0}^\infty \frac{\cos(2\pi(8k+1) x)}{(8k+1)^2} + \sum_{k=0}^\infty \frac{\cos(2\pi(8k+2) x)}{(8k+2)^2} + \cdots \right. \\ \,\,\,\, \left. + \sum_{k=0}^\infty \frac{\cos(2\pi(8k+7) x)}{(8k+7)^2} + \sum_{k=1}^\infty \frac{\cos(2\pi(8k) x)}{(8k)^2}\right)[/tex]
which reduces to the identity
[tex]\pi^2\left(\left(x-\dfrac12\right)^2-\dfrac{21}{256}\right) = \\\\ \cos(2\pi x) {S_1}' + \cos(4\pi x) {S_2}' + \cos(6\pi x) {S_3}' + \cos(8\pi x) {S_4}' \\\\ \,\,\,\, + \cos(10\pi x) {S_5}' + \cos(12\pi x) {S_6}' + \cos(14\pi x) {S_7}'[/tex]
Evaluating both sides at x for x ∈ {1/8, 3/8, 5/8, 7/8} and solving the system of equations yields the dependent solution
[tex]\begin{cases}{S_4}' = \dfrac{\pi^2}{256} \\\\ {S_1}' - {S_3}' - {S_5}' + {S_7}' = \dfrac{\pi^2}{8\sqrt 2}\end{cases}[/tex]
It turns out that
[tex]{S_1}' - {S_3}' - {S_5}' + {S_7}' = S_1 - S_3 - S_5 + S_7[/tex]
so we're done, and the sum's value is [tex]\boxed{\dfrac{\pi^2}{8\sqrt2}}[/tex].
![[tex]\large \rm \sum \limits_{n = 0}^ \infty \frac{( { - 1)}^{1 + 2 + 3 + \dots + N} }{(2n + 1 {)}^{2}](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/Fgk8Aqp2bSwmQdJ7IWnS7LRiW1nD0Akf.png)
Related Questions
3. For each triangle, find the length of the labeled side.
Answers
Answer:
see explanation for detailed analysis
Can somebody help me pls!
Answers
Answer: C
Step-by-step explanation:
Just look at a z-score table and multiply by 100.
-> (0.308538)(100) is about 30.85%
Señora Cruz will use four triangles on the door decor. How many square centimeters of paper will Señora Cruz use to create the triangles?
Triangle has bases as 3.7 and height as 6.8
Square has length as 7.5 and width as 3.9
Answers
Answer:
50.32 cm²
Step-by-step explanation:
The area of a triangle can be computed using the formula ...
A = 1/2bh
For a triangle with base 3.7 cm and height 6.8 cm, the area is ...
A = 1/2(3.7 cm)(6.8 cm) = 12.58 cm²
__
For four (4) triangles, Señora Cruz will need 4 times this area:
4 × (12.58 cm²) = 50.32 cm²
Señora Cruz will need 50.32 cm² of paper to create her door decor.
A card is picked from a standard deck of 52 cards. Determine the odds against and the odds in favor of selecting a red face card (king, queen, or jack).
Answers
6 red face cards
->in favour:
6/52
= 3/26
-> against:
52-6= 46
46/52
=23/26
jack had m math problems to complete during his vacation. he solved the same number of problems every day and finished them all in 5 days. how many problems did jack solve per day.
Answers
If Jack finish them all in 5 days, then he can solve m/5 math problem in just one day.
Word problems leading to quadratic equationFrom the given question, jack can only solve m math problems in 5 days, this can be expressed as:
m problems = 5 days
The number of questions he can solve per day is expressed as:
x = 1 day
Take the ratio
m/x = 5/1
5x = m
x = m/5 math problems
This shows that jack can solve m/5 math problem in just one day
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Find the mean of the data.
8,14,22,7,2,11,25,7,5,9
Answers
Answer:
11
Step-by-step explanation:
Given:
8,14,22,7,2,11,25,7,5,9
Solve:
Put in order:
2, 5, 7, 7, 8, 9, 11, 14, 22, 25
Note:
Mean-
Add up all data values to get the sumCount the number of values in your data setDivide the sum by the count2+ 5+7+7+8+9+11+ 14+22+25=110
110/10 = 11
Hence, the mean of the data is 11.
[RevyBreeze]
Answer:
The mean of the data given is 11
What is mean?
The mean is the arithmetic average of a set of given numbers. The median is the middle score in a set of given numbers. The mode is the most frequently occurring score in a set of given numbers.
Step-by-step explanation:
Have a great rest of your day
#TheWizzer
(pls give the person who answered before me braineist)
a thousand dollars is left in a bank savings account drawing 7% interest, compounded quarterly for 10 years. what is the balance at the end of that time
Answers
Work Shown:
A = P*(1+r/n)^(n*t)
A = 1000*(1+0.07/4)^(4*10)
A = 2001.59734318603
A = 2001.60
This assumes that you do not withdraw any of the money over the course of the 10 years. Also, the interest rate must stay the same at 7%.
Find the area of the following shape. (8 points)
what is the answer?
Answers
Answer:
72 square units
Step-by-step explanation:
Identify the heightIdentify the lengthMultiply those two togetherThat is your answerH = 9 units
L = 8 units
A = H × L
A = 9 × 8
A = 72 square units
A can of soda can be modeled as a right cylinder. Aubrey measures its height as 10.1
cm and its circumference as 14 cm. Find the volume of the can in cubic centimeters.
Round your answer to the nearest tenth if necessary.
Answers
The volume of the cylinder is the amount of space in the cylinder
The volume of the cylinder is 218.4 cubic centimeter
How to determine the volume of the cylinder?The given parameters are:
Circumference (C) = 14 cm
Height (h) = 10.1 cm
Start by calculating the radius (r) using:
[tex]C = 2\pi r[/tex]
So, we have:
[tex]2\pi r = 14[/tex]
Make r the subject
[tex]r =\frac 7\pi[/tex]
The volume of the cylinder is then calculated as:
[tex]V =\pi r^2h[/tex]
This gives
[tex]V =\pi * (\frac{7}{\pi})^2 * 14[/tex]
Evaluate
[tex]V =218.4[/tex]
Hence, the volume of the cylinder is 218.4 cubic centimeter
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the equation is :
answer x:
Answers
Answer:
A) x would be 21 if i interpreted it right
Step-by-step explanation:
4x - 11 = 73
i think anyways
4x = 73 + 11
4x = 84
x = 21
i d k what B means?
Help solve for “q”
—————————————
Answers
Digram:-
[tex] \\ [/tex]
[tex]\setlength{\unitlength}{1 cm}\begin{picture}(0,0)\thicklines\put(5,1){\vector(1,0){4}}\put(5,1){\vector(-1,0){4}}\put(5,1){\vector(1,1){3}}\put(2,2){$\underline{\boxed{\large\sf a + b = 180^{\circ}}$}}\put(4.5,1.3){$\sf a^{\circ}$}\put(5.7,1.3){$\sf b^{\circ}$}\end{picture}[/tex]
[tex] \\ [/tex]
STEP :-
[tex] \dashrightarrow \tt(4q - 1) {}^{ \circ} + {117}^{ \circ} = 18 {0}^{ \circ} [/tex]
{Linear pair}
[tex] \\ \\ [/tex]
[tex] \dashrightarrow \tt(4q - 1) {}^{ \circ}= 18 {0}^{ \circ} - {117}^{ \circ}[/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt(4q - 1) {}^{ \circ}=63^{ \circ}[/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt4q - 1{}^{ \circ}=63^{ \circ}[/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt4q =63^{ \circ} + 1{}^{ \circ}[/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt4q =64{}^{ \circ}[/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt \: q = \dfrac{64}{4}^{ \circ}[/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt \: q = \dfrac{16 \times 4}{4}^{ \circ}[/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt \: q = \dfrac{16 \times \cancel4}{\cancel4}^{ \circ}[/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt \: q = \dfrac{16}{1}[/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \bf q = 16 \degree[/tex]
[tex] \\ \\ [/tex]
Verification:
[tex] \\ [/tex]
[tex] \dashrightarrow \tt(4 \times 16- 1) {}^{ \circ} + {117}^{ \circ} = 18 {0}^{ \circ} [/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt(64- 1) {}^{ \circ} + {117}^{ \circ} = 18 {0}^{ \circ} [/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt63^{ \circ} + {117}^{ \circ} = 18 {0}^{ \circ} [/tex]
[tex] \\ [/tex]
[tex] \dashrightarrow \tt180^{ \circ} = 18 {0}^{ \circ} [/tex]
[tex] \\ [/tex]
LHS = RHS
HENCE VERIFIED!
Answer:
Value of [tex]\sf\purple{q\: = \:16.}[/tex]
Step-by-step explanation:
[tex]\rightarrow[/tex]As we know that,
Sum all angles that lie on a straight line = [tex]\sf\blue{180°}[/tex]
So,
[tex]\rightarrow[/tex] [tex]\sf{(4q-1)°+ 117°\: = \:180°}[/tex]
[tex]\rightarrow[/tex] [tex]\sf{(4q-1)\: = \:180-117}[/tex]
[tex]\rightarrow[/tex] [tex]\sf{(4q-1)\: = \:63}[/tex]
[tex]\rightarrow[/tex] [tex]\sf{4q\: = \:63+1}[/tex]
[tex]\rightarrow[/tex] [tex]\sf{q\: = \:\frac{64}{4}}[/tex]
[tex]\rightarrow[/tex] [tex]\sf{q\: = \:16}[/tex]
Thus, [tex]\sf\purple{q\: = \:16.}[/tex]
_________________________________
Hope it helps you:)
4
Find the perimeter of a
Square with a side
length of 7 meters.
Answers
Answer: P=28 meters
Step-by-step explanation:
[tex]P=4[/tex] × a ⇒ a is the side length
[tex]P=4[/tex] × [tex]7[/tex]
[tex]P=28[/tex]
Answer:
28 m
Step-by-step explanation:
Given
Side length = 7 mPerimeter of a square
4 x Side length4 x 728 mHelp help math math math math math
Answers
Answer:
A
Step-by-step explanation:
You can think about it as an equation without the inequality:
y = 5 - x OR y = -x + 5
Slope = -1
Y-intercept = 5
Graph B is a horizontal line with a slope of zero and y-intercept of 2. Graph A is the only one that fits the above parameters.
Hope this helps!
Answer:
a
Step-by-step explanation:
What is the approximate volume of a cone with a height of 9 ft and radius of 3 ft? Use 3.14 to approximate pi, and express your final answer to the nearest hundredth Enter your answer as a decimal in the box. ft3
Answers
84.78 ft3
Steps:
Take note.
V = Volume
r = radius
h = height
V = (Pi) * (r^2 ) * (h/3)
V = (3.14) * (3^2) * (9/3)
V = (3.14) * (9) * (3)
V = 84.78
help for brainliest and 10 points
Answers
Answer:
-3
Step-by-step explanation:
[tex]0.002 = 2 \times {10}^{ - 3} \\ \implies \: x = - 3[/tex]
9x9/16+12 whats the answer
Answers
Answer:
17.0625
Step-by-step explanation:
Due to order of operations the division and multiplication get done first and the we add the 12
Answer:
2.89285714
Step-by-step explanation:
9x9/16+12
81/16+12
5.0625 + 12 = 17.0625
Hope this helps :)
A Ferris wheel is 20 meters in diameter and completes 1 full revolution in 8 minutes.
A round Ferris wheel
A Ferris wheel is 20 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 8 minutes. The function h(t) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.
a. Find the amplitude, midline, and period of h(t).
Enter the exact answers.
Amplitude: A= meters
Midline: h= meters
Period: P= minutes
b. Assume that a person has just boarded the Ferris wheel from the platform and that the Ferris wheel starts spinning at time t=0. Find a formula for the height function h(t).
Hints:
What is the value of h(0)?
Is this the maximum value of h(t), the minimum value of h(t), or a value between the two?
The function sin(t) has a value between its maximum and minimum at t=0 , so can h(t) be a straight sine function?
The function cos(t) has its maximum at t=0, so can h(t) be a straight cosine function?
c. If the Ferris wheel continues to turn, how high off the ground is a person after 30 minutes?
Answers
Given that the Ferris diameter is 20 meters, with a rate of rotation of 1 turn in 8 minutes, we have;
a. Amplitude: A = 10 meters
Midline: h = 11 meters
Period, P = 8 minutes
b. h(t) = 10•cos((π/4)•t + π) + 11
c. 11 meters
How can the Ferris wheel be evaluated?The amplitude is the same as the radius of the Ferris wheel,
The radius of the Ferris wheel = 20 ÷ 2 = 10
Therefore;
Amplitude: A = 10 meters[tex]the \: midline \: = \frac{max \: height \: + min \: height}{2} [/tex]
Therefore;
[tex]midline \: = \frac{10 + 10 + 1 + 1}{2} = 11[/tex]
Midline: h = 11 metersThe period is the time to complete one rotation, therefore;
Period, P = 8 minutesb. h(t) = A•cos(B•t + C) + h
Where;
B = 2•π/P
When t = 0, h(t) = 1
Which gives;
h(0) = 1 = 10 × cos(B×0 + C) + 11
-10/10 = -1 = cos(C)
C = arcos(-1) = π
Therefore;
h(t) = 10•cos((π/4)•t + π) + 11h(0) = 1
h(0) is the minimum value of h(t)
h(t) cannot be a straight sine function because of the vertical shifth(t) cannot be a straight cosine function because at t = 0, is the minimum pointc. After 30 minutes, we have;
h(30) = 10•cos((π/4)×30 + π) + 11 = 11
The height of a person after 30 minutes is 11 metersLearn more about the Ferris wheel here:
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A perfect score on a test with 25 questions is 100. Each question is worth the same number of points. How many points is each question on the test worth
Answers
Answer:
4
Step-by-step explanation:
100 divided by 25 equals 4.
1. For each diagram below, find the value of x
Answers
b. x + 110 = 140
x = 140 - 110
x = 30
c. 3x + 7 = x + 35
2x = 28
Divide through by 2
x = 14
solve this question plss on number line
Answers
Answer:
3. 13/36, 14/36, 15/36, 16/36, 17/36
4.
-0.5 -0.25 0 0.25 0.5
5. 0.05
3.
between 1/2 and 1/3five rational numbers:
0.35, 0.4, 0.42, 0.45, 0.46
4. Given in the graph below.
5. (-2/5 + 1/2)/2 = 0.05
Please the answer ... Integral
Answers
Answer:
[tex]\frac{dx^{2} (x+1)S^{2} }{2(x^{2} +6x+3)^{2} }+ C[/tex]
Step-by-step explanation:
Find the area of sector RST Enter your answer in terms of a fraction of it and rounded to the nearest
hundredth.
Answers
Fort nite battle pass is 8 dollars
?A bag contains red, blue, and green candies. Benjamin pour
out a handful and counted 10 red, 6 blue, and 14 green
candies. According to these ratios, if the bag contains a total of
400 candies, about how many of them are blue?
Answers
Find the missing information for the triangle.
*not drawn to scale
• Make sure to find the missing angle measure and the 2 missing side
lengths.
Answers
missing angle:
180° - 90° - 30°
180° - 120°
60°
missing sides:
(a)
[tex]\rightarrow \sf tan(x)= \dfrac{opposite}{adjacent}[/tex]
[tex]\rightarrow \sf tan(30)= \dfrac{4}{adjacent}[/tex]
[tex]\rightarrow \sf adjacent= \dfrac{4}{tan(30)}[/tex]
[tex]\rightarrow \sf adjacent= 4\sqrt{3}[/tex]
[tex]\rightarrow \sf adjacent= 6.93 \ cm[/tex]
(b)
[tex]\sf \rightarrow sin(x)= \dfrac{opposite}{hypotensue}[/tex]
[tex]\sf \rightarrow sin(30)= \dfrac{4}{hypotensue}[/tex]
[tex]\sf \rightarrow hypotensue= \dfrac{4}{ sin(30)}[/tex]
[tex]\sf \rightarrow hypotensue= 8 \ cm[/tex]
Answer:
m∠X = 60°
BX = 8 cm
BM = 4√3 cm
Step-by-step explanation:
The sum of the interior angles of a triangle is 180°
Given:
m∠B = 30°m∠M = 90°⇒ m∠B + m∠M + m∠X = 180°
⇒ 30° + 90° + m∠X = 180°
⇒ 120° + m∠X = 180°
⇒ m∠X = 180° - 120°
⇒ m∠X = 60°
Using the sine rule to find the side lengths:
[tex]\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]
(where A, B and C are the angles, and a, b and c are the sides opposites the angles)
Given:
m∠X = 60°m∠B = 30°m∠M = 90°MX = 4 cm[tex]\implies \dfrac{4}{\sin 30\textdegree}=\dfrac{BX}{\sin 90\textdegree}=\dfrac{BM}{\sin 60\textdegree}[/tex]
[tex]\implies BX=\sin 90\textdegree \cdot\dfrac{4}{\sin 30\textdegree}[/tex]
[tex]=1 \cdot \dfrac{4}{\frac12}[/tex]
[tex]=1 \cdot 4 \cdot 2[/tex]
[tex]=8 \textsf{ cm}[/tex]
[tex]\implies BM=\sin 60\textdegree \cdot\dfrac{4}{\sin 30\textdegree}[/tex]
[tex]=\dfrac{\sqrt{3}}{2}\cdot \dfrac{4}{\frac12}[/tex]
[tex]=\dfrac{\sqrt{3}}{2}\cdot 4 \cdot 2[/tex]
[tex]=4\sqrt{3} \textsf{ cm}[/tex]
WILL GIVE EXTRA POINTS FOR ANSWER ⭐️⭐️!! PLEASE EXPLAIN IF POSSIBLE
Answers
Answer:
B. (-3, 10)
Step-by-step explanation:
I am going to graph the given equation. I then will see which of the points given are within the required area.
-> See attached.
-> I have explained in the image more in-depth as well.
y=5/2x-9 find the y intercept
Answers
Answer:
(0,-9) You have to substitute 0 for x and solve for y
How to solve x + y[tex]\frac{dy}{dx}[/tex] = 0
Given that solution goes to (2,0) Neither x nor y can exceed 2
Answers
Answer:
The answer is {D}
Step-by-step explanation:
The temperature at 1:00 p.m. on Tuesday was -13°C. There was an increase of 6º per
hour starting at 1:00 p.m. Which of the following best represents the Celsius
temperature n hours after 1:00 p.m. on Tuesday?
A. -13 + bn
B. -13 - 6n
C. -13n + 6
D. -13n - 6
Answers
At 1.00Pm the temperature was -13°C
No of hours be nIncrease rate=6°C/hourSo
The equation is
y=6n+(-13)y=6n-13y=-13+6nCan somebody please help with this, I have been stuck on it for a while
Answers
Answer:
$2821.50
Step-by-step explanation:
value = 2700 (deposit) x 0.003 (rate) x 15 (time) + 2700
[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$2700\\ r=rate\to 0.3\%\to \frac{0.3}{100}\dotfill &0.003\\ t=years\dotfill &15 \end{cases} \\\\\\ A=2700[1+(0.003)(15)]\implies A=2700(1.045)\implies A=2821.5[/tex]
16+32 as a product of two factors using gif and distributive property
Answers
16 + 32 = 16 x 1 + 16 x 2 = 16 x (1 + 2)