Rewrite the factorial parts of the summand as
[tex]\dfrac{(2n+1)!}{n!(n+2)!} = \dfrac{(2n+1)(2n!)}{(n+2)(n+1)(n!)^2} = \dfrac{2n+1}{(n+2)(n+1)} \dbinom{2n}n[/tex]
where [tex]\binom nk[/tex] is the binomial coefficient, and [tex]\binom{2n}n[/tex] are the so-called central binomial coefficients.
Expand the rational expression into partial fractions:
[tex]\dfrac{2n+1}{(n+2)(n+1)} = \dfrac3{n+2} - \dfrac1{n+1}[/tex]
Pull out a constant factor and collect the exponential terms.
[tex]\dfrac{(-1)^{n+1}}{4^{2n+3}} = -\dfrac1{64} \left(-\dfrac1{16}\right)^n[/tex]
The sum we want is now
[tex]\displaystyle \frac{13}8 - \frac1{64} \sum_{n=0}^\infty \binom{2n}n \left(\frac3{n+2} - \frac1{n+1}\right) \left(-\frac1{16}\right)^n[/tex]
Let f(x) and g(x) be functions with power series expansions
[tex]\displaystyle f(x) = \sum_{n=0}^\infty \binom{2n}n \frac{x^n}{n+1}[/tex]
[tex]\displaystyle g(x) = \sum_{n=0}^\infty \binom{2n}n \frac{x^n}{n+2}[/tex]
and recall the well-known binomial series
[tex]\displaystyle \dfrac1{\sqrt{1-4x}} = \sum_{n=0}^\infty \binom{2n}n x^n[/tex]
which converges for |x| < 1/4.
Integrating both sides yields
[tex]\displaystyle \int \frac{dx}{\sqrt{1-4x}} = \int \sum_{n=0}^\infty \binom{2n}n x^n \, dx[/tex]
[tex]\displaystyle -\frac12 \sqrt{1-4x} = C_1 + \sum_{n=0}^\infty \binom{2n}n \frac{x^{n+1}}{n+1}[/tex]
Taking x = 0 on both sides, it follows that C₁ = -1/2. We then see that
[tex]\displaystyle f(x) = \frac{1-\sqrt{1-4x}}{2x}[/tex]
Step back and multiply both sides of the binomial series identity by x, then integrate. This yields
[tex]\displaystyle \int \frac x{\sqrt{1-4x}} \, dx = \int \sum_{n=0}^\infty \binom{2n}n x^{n+1} \, dx[/tex]
[tex]\displaystyle -\frac1{12} \sqrt{1-4x} (1 + 2x) = C_2 + C_1 x + \sum_{n=0}^\infty \binom{2n}n \frac{x^{n+2}}{n+2}[/tex]
Taking x = 0 again points to C₂ = -1/12. Hence
[tex]\displaystyle g(x) = \frac{1 - \sqrt{1-4x}(1+2x)}{12x^2}[/tex]
Then the value of the sum we want is
[tex]\displaystyle \frac{13}8 - \frac1{64} \left(3g\left(-\frac1{16}\right) - f\left(\frac1{16}\right)\right) = \frac{1+\sqrt5}2 = \boxed{\phi}[/tex]
where ɸ ≈ 1.618 is the golden ratio.
Solve for x.
√10x−29=x−2
Enter your answers in the boxes.
Answer:
x = 11, 3
Step-by-step explanation:
√(10x - 29) = x - 2
(√(10x - 29))² = (x - 2)²
10x - 29 = x² - 4x + 4
-10x + 29 -10x + 29
x² - 14x + 33
(x - 11) (x - 3)
x - 11 = 0; x - 3 = 0
+ 11 +11 +3 +3
x = 11 x = 3
I hope this helps!
The body temperatures of a group of healthy adults have a bell-shaped distribution with a mean of 98.27°F and a standard
deviation of 0.54°F. Using the empirical rule, find each approximate percentage below.
a. What is the approximate percentage of healthy adults with body temperatures within 1 standard deviation of the mean, or
between 97.73 °F and 98.81°F?
Answer: follow this you'll be able to solve it
Step-by-step explanation: mean = 98.11F
standard deviation = 0.56F
99.79 – 98.11 = 1.68 = 3 standard deviations
96.43 – 98.11 = –1.68 = –3 standard deviations
96.43F and 99.79F are 3 standard deviations from the mean 98.11F.
By the empirical rule we know that 99.7% of the data lies within 3 standard deviation of the mean.
Approximately 68% of healthy adults in this group have body temperatures within 1 standard of the mean, or between 97.55F and 98.67F.
What is the quotient when 7.8 is divided by 6 ?
Answer:
the anser is 1.3
have a nice day!
You have two cubes that
you fill with water
to make ice cubes. The first cube has a side
length of 3 units. The second cube has a
side length of 5 units. Write an
expression
that you can use to find the total volume of
ice you can make.
Answer:
V = 3^3 = 5^3
Volume of each cube is side^3. To find the volume of both cubes just add the two volumes together.
Step-by-step explanation: Your welcome.
Answer:
total volume = 152 units³
given:
first cube side: 3 unitssecond cube side: 5 unitsvolume of cube: side³using this formula:
total volume = first cube volume + second cube volume
total volume = 3³ + 5³
total volume = 27 + 125
total volume = 152 units³
what is the area of a triangle with vertices at (0,0), (0,4) and (4,5)?
Answer:
8
Step-by-step explanation:
Let us label these coordinates for our own ease:
A(0,0)
B(0,4)
C(4,5)
Area= [Ax(By-Cy)+Bx(Cy-Ay)+Cx(Ay-By)]÷2
Area = [0(4-5)+0(5-0)+4(0-4)]÷2
Area= [0+0-16]÷2
Area = -8
But since the value can not be negative we have consider a modulus in the formula, thus the value is 8
Students are given 3 minutes to complete each multiple-choice question on a test and 8 minutes for each free-response question. There are 15 questions on the test and the students have been given 55 minutes to complete it.
A Table titled Test Time, showing Number of Questions, Time per Item in minutes, and Total Time in minutes. The first row shows Multiple Choice, with m, 3, and 3 m. The second row shows Free Response, with 15 minus m, 8, and x. The third row shows Total, with 15, blank, and 55.
Which value could replace x in the table?
Answer:
8(15-m)
Step-by-step explanation:
just took the test
You can work a total of no more than 10 hours each week at your iwo jobs. House cleaning pays $5 per hour and
your sales job pays $8 per hour. You need to earn at least $56 each week to pay your bills. Write a system of
inequalities that show the various numbers of hours you can work at each job.
I'm not 100% sure but it's not u
Fred completed 10 math problems. This is 40% of the number of math problems he has to do. How many math problems must he do?
Group of answer choices
40
30
25
15
Answer:
25
Step-by-step explanation:
we find how many problems 1% is first by solving 10/40, which gives us .25
then, we multiply by 100, which results in 25.
i could try to explain better if you'd like
Do these ratios form a proportion?
$30 per 46 pounds
$15 per 23 pounds
Answer: Yes!
Step-by-step explanation:
a proportion is when to ratios are equivalent to each other and that they are set equal to each other
If you see, $30 is divided by 2 to get $15. Same goes for the pounds, 46 is divided by 2 to get 23. They follow the same proportion
Sarah normally earns £600 each month. In December she is given a bonus of 1/5 of her normal pay. How much will she now earn this month?
Answer:
She will earn 720 in December(Sorry, can't use the dollar sign.)
Step-by-step explanation:
1/5 of 600 is 120, so you want to add 120 to 600, which would be 720. Hope this helps!
Write an equation that can be described by the statement: y is equal to 1/3 of x.
Enter the correct answer in the box.
Answer: [tex]y=\frac{1}{3} x[/tex]
4/7×4/5×7/9
Write your answer in fraction in simplest form
Answer:
16/45 for fraction, or 0.35 for decimal
Step-by-step explanation:
4/7 x 4/5 × 7/9
reduce numbers with greatest common factor 7
4 × 4/5 × 1/9
calculate the product = 4 × 4 × 1 = 16
5 × 9 = 45
16/45 or 0.35
The water level in a tank measures 15 inches and is decreasing 0.5 inches every minute. How many minutes, x, will it take for the water level to measure 9 inches deep?
HELP!!! 100 POINTS ANSWER FAST
Answer:
(x - 4)^2 = 9Step-by-step explanation:
Completing the square
x^2 - 8x + 7 = 0x^2 - 2*4x + 16 - 9 = 0(x^2 - 2*4x + 4^2) = 9 (x - 4)^2 = 9The blanks are
- 4 and 9Answer:
(x^2-4)^2 = 9
Step-by-step explanation:
First, isolate the constant:
x^2-8x = -7
Then, since (a-b)^2 = a^2-2ab+b^2, you should add 16(4^2) to both sides:
x^2-8x+16 = 9
Now, we can rewrite the right-hand side as (a-b)^2:
(x-4)^2 = 9
The sum of teo numbers is 459909 . One number 400019. Find the other number?
Answer:
59890
you just subtract
459909-400019
Step-by-step explanation:
please pass this on... ( #helpsavelives )
Who ever is reading this:
I love u.
U matter.
The world needs u
hang in there ik it may be bad but u deserve the world <3
ur beautiful no matter ur shape, size, color, gender.. anything
don't give up i need u to live.
i wish i could take everyones problems so yall wouldn't have to have em but i cant so just know ily and if anyone needs to talk u can talk to me
Pls pass this on. Everyone deserves to know this. ♥️♥️
just copy and paste its not hard pls this could save someones life
Answer:
59890
Step-by-step explanation:
459909-400019
Let g(x) = 2x and h(x) = x2 + 4.
Evaluate (h ∘ g)(−2).
20
−12
−16
16
Answer:
Option C, 20 is the answer.
Step-by-step explanation:
Let g(x) = 2x and h(x) = x² + 4
We have to evaluate (h ο g) ( -2 )
Since ( h ο g )x = h [ g(x) ]
= ( 2x)² + 4
= 4x² + 4
( h ο g ) ( -2 ) = 4 ( -2 )² + 4
= 4 × 4 + 4
= 16 + 4
= 20
Hope it helps!!!Brainliest pls!!!represent 20 degree 30' in radian
Answer:
20 D is 72%
Step-by-step explanation:
Which of the following are factor pairs for 72? You will select two.
2 x 36
3 x 18
6 x 12
8 x 8
Answer:
The two factor pairs for 72 are the following:
2 × 366 × 12Step-by-step explanation:
Of you take the answer to each equation only these two equal 72, leaving them as factors.
3 × 18 is equal to 54
8 × 8 is equal to 65
Which leaves them being the wrong answer
Colin’s piano lesson is 45 minutes, or Three-fourths of an hour, long. If he practices 6 different songs and spends an equal amount of time on each song, what fraction of an hour does he spend practicing each song?
Answer:
Colin spends 1/8 of an hour practicing each song
Step-by-step explanation:
Colin's piano lesson is 3/4 of an hour, long.
In that period, he practices 6 different songs, spending an equal amount of time on each one.
This means that each song gets the sixth part of the lesson duration. The fraction that expresses that amount is:
[tex]\displaystyle \frac{\frac{3}{4}}{6}=\frac{3}{24}=\frac{1}{8}[/tex]
Colin spends 1/8 of an hour practicing each song
Answer:
1/8
Step-by-step explanation:
help please
find the value
Answer:
t = 9 cm
Step-by-step explanation:
Use Pythagoras' Theorem: a² + b² = c²
(where a and b are the legs and c is the hypotenuse of a right triangle)
Given:
a = 12b = tc = 15a² + b² = c²⇒ 12² + t² = 15²
⇒ 144 + t² = 225
⇒ t² = 225 - 144
⇒ t² = 81
⇒ t = √81
⇒ t = ±9
⇒ t = 9 only as length is always positive
Solution:
Pythagoras theorem can only be used in:
Right trianglesScalene right trianglesIsosceles right triangles.Since this is a scalene right triangle, we can use Pythagoras theorem to solve the missing length of the triangle.
=> The formula for Pythagoras theorem is a² + b² = c².Finding a, b, and c:
Let "a" be 12 cm and "b" be "t" cm."c" is the largest side of the triangle. This means that "c" is 15 cm, as it is the largest side of the triangle.Setting up the equation:
a² + b² = c²=> 12² + t² = 15²Simplifying the squares:
=> (12)(12) + t² = (15)(15)Subtracting (12)(12) both sides:
=> t² = -(12)(12) + (15)(15)=> t² = -144 + 225=> t² = 81Taking a square root both sides:
=> √t² = √81=> t = ±9Since the length cannot be negative, the value of t must be positive.
The final answer is...
t = 9is -7 a solution to the equation 8(1/4x+3/4)-3+17?
yes
no
Elsa earns 5140 for working 20 hours
1) What is Elsa's rate of pay for each hour she works?
A) 5650 per hour
B) $7.00 per hour
$7.25 per hour
D) $2.00 per hour
Answer:
$2.57
Step-by-step explanation:
divide 5,140 by 20,
5,140/20=257=2.57
Hello can someone please answer this question i will mark you the Brainliest
Answer:
A) 310 cm²
Step-by-step explanation:
The surface area is the area of the two-dimensional surfaces of this three-dimensional figure.
Start by multiplying each 5x4 rectangle. There are four of them that are visible from the front.
4(5 · 4) 4(20) = 80 cm²Now, find the surface area of the large rectangle in the back and the left side of the figure.
2(10 · 4) = 80 cm²Find the surface area of the top of this figure.
10 · 5 = 50 cm²5 · 5 = 25 cm²Find the surface area of the bottom of this figure (it's the same as the area of the top).
50 + 25 = 75 cm²Add all of the surface areas together to find the total surface area of the figure.
80 + 80 + 75 + 75 = 310 cm²State if three numbers can be the measure of a triangle.
12, 12, 20
10,3,7
7,7,11
24, 9, 12
Given that r = 9.2 m and = 17", work out AB rounded to 3 SF
Answer:
2.813 m
Step-by-step explanation:
x = 9.2 θ = 17 AB=?
Looking at our angle we have an adjacent side & need to find the opposite side.
So tan θ = opp/adj
tan 17 = AB/9.2
AB = 9.2tan17
AB = 2.813 m
If x = -12, then solve:
-X =
(Please help me need answer by 1pm)
Answer:
-x = 12
Step-by-step explanation:
x = -12
if you multiply both sides of the equation by -1, you get
-1x = -12 * -1
-x = 12
Answer:
12
Step-by-step explanation:
So what we know is that x is equal to negative 12. To make this easier you can say that -x is the same as -1 times x like shown below,
-1 × [tex]x[/tex] is the same as -X
We can replace the x with -12 and a negative times a negative is a positive so the answer would be 12.
If you want me to explain it more let me know, but hopefully this helps and have a great day!
what is 15x1300-1000+14=?
Answer:
18514 if im wrong then sorry but i dont think its wrong
Step-by-step explanation:
one bath of cookies calls for 5 cups of flour for every 3 cups of chocolate chips how many cups of flour should be used for 27 cups of chocolate chips
Answer:
16.2
Step-by-step explanation:
0.6 * 27
Hope it helps!!!Brainliest pls!!!Evaluate:
[tex] \\ { \displaystyle{ \rm \lim_{x \to1} \left \{ \frac{ {x}^{4 } - {3x}^{2} + 2 }{ {x}^{3} - {5x}^{2} + 3x + 1} \right \} }} [/tex]
[tex] \: [/tex]
Don't Spam
Answer:
To Evaluate:
[tex] \\ { \displaystyle{ \rm \lim_{x \to1} \left \{ \frac{ {x}^{4 } - {3x}^{2} + 2 }{ {x}^{3} - {5x}^{2} + 3x + 1} \right \} }}\\ \\[/tex]
Using L'Hospital's Rule
Let f(x) and g(x) be two common functions which are differentiable on an open Interval , at a point a where,
[tex]\\{ \displaystyle{ \lim_{x \to a}{ \rm{f(x)} = { \displaystyle{ \lim_{x \to a}}} \: { \rm{g(x)}} = 0 \: or \: \pm \infty}}}\\ \\[/tex]
Then,
[tex]\\{ \displaystyle{ \lim_{x \to a} \frac{{ \rm{f (x)}}}{{ \rm{g(x)}}} = { \displaystyle{ \lim_{x \to a }}} \frac{{ \rm{f '(x)}}}{{ \rm{g'(x)}}}}}\\ \\[/tex]
As x to 0 , we have:
[tex]\\{ \displaystyle{ \lim_{x \to 1}}} \left( \large\rm\frac{ {x}^{4} - {3x}^{2} + 2 }{ {x}^{3} - {5x}^{2} + 3x + 1} \right) = \frac{0}{0}\\ \\[/tex]
Therefore,
[tex]\\{ \large{ \displaystyle{ \lim_{x \to1}}} \left( \large \rm\frac{ {x}^{4} - {3x}^{2} + 2}{ {x}^{3} - {5x}^{2} + 3x + 1 } \right) = { \displaystyle{ \lim_{x \to1} { \small{\frac{ {4x}^{3} - 6x}{ {3x}^{2} - 10x + 3 }}} = { \small{\frac{4 - 6}{3 - 10 + 3} }} = - \frac{2}{ - 4} = - \frac{1}{2} }}}\\ \\[/tex]
Hence,
[tex]\\{ \displaystyle{ \lim_{x \to1}{\rm{ \left( \frac{ {x}^{4} - 3 {x}^{2} + 2 }{ {x}^{3} - 5 {x}^{2} + 3x + 1 } \right) = { \boxed{ \red{ - \frac{1}{2} }}}}}}}\\ \\[/tex]
Without using L'Hopital's rule:
Factorize the numerator.
[tex]x^4 - 3x^2 + 2 = (x^2 - 2) (x^2 - 1) = (x^2 - 2) (x + 1) (x - 1)[/tex]
Factorizing cubics is usually a bit trickier. But we know x = 1 makes the denominator vanish, so x - 1 is a factor of the cubic. So for some constants a and b,
[tex]x^3 - 5x^2 + 3x + 1 = (x - 1)^3 + a(x - 1)^2 + b(x - 1)[/tex]
Expanding the right side gives
[tex]x^3 - 5x^2 + 3x + 1 = x^3 + (a - 3)x^2 + (-2a + b + 3)x + a - b - 1[/tex]
Then
[tex]a - 3 = -5 \implies a = -2[/tex]
[tex]-2a + b + 3 = 3 \implies b = -4[/tex]
So we rewrite the limit as
[tex]\displaystyle \lim_{x\to1} \frac{x^4 - 3x^2 + 2}{x^3 - 5x^2 + 3x + 1} = \lim_{x\to1} \frac{(x^2 - 2) (x + 1) (x - 1)}{(x-1)^3 - 2(x-1)^2 - 4(x-1)}[/tex]
x is approaching 1 so x ≠ 1 and we can cancel out factors of x - 1 :
[tex]\displaystyle \lim_{x\to1} \frac{x^4 - 3x^2 + 2}{x^3 - 5x^2 + 3x + 1} = \lim_{x\to1} \frac{(x^2 - 2) (x + 1)}{(x-1)^2 - 2(x-1) - 4}[/tex]
The simplified limand is continuous at x = 1, so we can now evaluate the limit by direct substitution.
[tex]\displaystyle \lim_{x\to1} \frac{x^4 - 3x^2 + 2}{x^3 - 5x^2 + 3x + 1} = \frac{(1^2 - 2) (1 + 1)}{(1-1)^2 - 2(1-1) - 4} = -\frac24 = \boxed{-\frac12}[/tex]
You bought a notebook for $2 and 6 packs of crayons. If you spent a total of $26, how much was each pack of crayons? *
Answer:
$4
Step-by-step explanation:
It was 4 dollars because you bought one notebook worth two dollars and six-packs of crayons. 26-2 is 24, and 6x4 is 24, so it is worth four dollars.