(a) Decompose the summand into partial fractions:
[tex]\dfrac5{n^2+n} = \dfrac5{n(n+1)} = \dfrac an+\dfrac b{n+1}[/tex]
[tex]\implies 5=a(n+1)+bn=(a+b)n+a[/tex]
[tex]\implies a+b=0\text{ and }a=5 \implies b=-5[/tex]
[tex]\implies\displaystyle\sum_{n=1}^\infty\frac5{n^2+n} = 5\sum_{n=1}^\infty\left(\frac1n-\frac1{n+1}\right)[/tex]
The n-th partial sum for the series is
[tex]S_n = 5\displaystyle\sum_{k=1}^n\left(\frac1k-\frac1{k+1}\right)[/tex]
which can be simplified significantly by examinging consective terms in the sum:
[tex]\displaystyle S_n = 5\left(1-\frac12\right) + 5\left(\frac12-\frac13\right) + 5\left(\frac13-\frac14\right) + \cdots + 5\left(\frac1{n-1}-\frac1n\right) + 5\left(\frac1n-\frac1{n+1}\right)[/tex]
[tex]\implies S_n = \boxed{5\left(1-\dfrac1{n+1}\right)}[/tex]
(b) Using the result of (a), you then get
[tex]\displaystyle\sum_{n=1}^\infty\frac5{n^2+n} = \lim_{n\to\infty}\boxed{5\left(1-\frac1{n+1}\right)} = \boxed{5}[/tex]
(c) As shown in (a), the partial sum is simplified because of the reasons given in options A and D, and the result of (b) says that B is also correct.
Answer:
Part a. [tex]\displaystyle S_n = 5 - \frac{5}{n + 1}[/tex]
Part b. [tex]\displaystyle \lim_{n \to \infty} (5 - \frac{5}{n + 1}) = 5[/tex]
Part c. A, B, and D
General Formulas and Concepts:
Algebra I
Terms/CoefficientsFactoringPre-Calculus
Partial Fraction DecompositionCalculus
Limits
Limit Rule [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to c} x = c[/tex]Limit Property [Addition/Subtraction]: [tex]\displaystyle \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)[/tex]Sequences
Series
Definition of a convergent or divergent seriesTelescoping Series: [tex]\displaystyle \sum^\infty_{n = 1} (b_n - b_{n + 1}) = (b_1 - b_2) + (b_2 - b_3) + (b_3 - b_4) + ... + (b_n - b_{n + 1}) + ...[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \sum^\infty_{n = 1} \frac{5}{n^2 + n}[/tex]
Step 2: Rewrite Sum
Factor: [tex]\displaystyle \sum^\infty_{n = 1} \frac{5}{n^2 + n} = \sum^\infty_{n = 1} \frac{5}{n(n + 1)}[/tex]Break up [Partial Fraction Decomposition]: [tex]\displaystyle \frac{5}{n(n + 1)} = \frac{A}{n} + \frac{B}{n + 1}[/tex]Simplify [Common Denominator]: [tex]\displaystyle 5 = A(n + 1) + Bn[/tex][Decomp] Substitute in n = 0: [tex]\displaystyle 5 = A(0 + 1) + B(0)[/tex]Simplify: [tex]\displaystyle 5 = A[/tex][Decomp] Substitute in n = -1: [tex]\displaystyle 5 = A(-1 + 1) + B(-1)[/tex]Simplify: [tex]\displaystyle 5 = -B[/tex]Solve: [tex]\displaystyle B = -5[/tex][Decomp] Substitute in variables: [tex]\displaystyle \frac{5}{n(n + 1)} = \frac{5}{n} + \frac{-5}{n + 1}[/tex]Simplify: [tex]\displaystyle \frac{5}{n(n + 1)} = \frac{5}{n} - \frac{5}{n + 1}[/tex]Substitute in decomp [Sum]: [tex]\displaystyle \sum^\infty_{n = 1} \frac{5}{n^2 + n} = \sum^\infty_{n = 1} \bigg( \frac{5}{n} - \frac{5}{n + 1} \bigg)[/tex]Step 3: Find Sum
Find Sₙ terms: [tex]\displaystyle \sum^\infty_{n = 1} \bigg( \frac{5}{n} - \frac{5}{n + 1} \bigg) = (5 - \frac{5}{2}) + (\frac{5}{2} - \frac{5}{3}) + (\frac{5}{3} - \frac{5}{4}) + (\frac{5}{4} - 1) + ... + ( \frac{5}{n} - \frac{5}{n + 1}) + ...[/tex]Find general Sₙ formula: [tex]\displaystyle S_n = 5 - \frac{5}{n + 1}[/tex]Find Sum [Take limit]: [tex]\displaystyle \sum^\infty_{n = 1} \frac{5}{n^2 + n} = \lim_{n \to \infty} S_n[/tex]Evaluate limit [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle \displaystyle \sum^\infty_{n = 1} \frac{5}{n^2 + n} = 5 + 0[/tex]Simplify: [tex]\displaystyle \sum^\infty_{n = 1} \frac{5}{n^2 + n} = 5[/tex]∴ the sum converges by the Telescoping Series.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Convergence Tests (BC Only)
Book: College Calculus 10e
What is the vertex of the graph of y = 2(x + 5)2 - 2?
Answer:
The vertex is (-5, -2)
Step-by-step explanation:
y = 2(x + 5)^2 - 2
The vertex form of a parabola is
y =a(x-h)^2 +k where (h,k) is the vertex
y = 2(x - -5)^2 - 2
The vertex is (-5, -2)
there are 10 crates of eggs stacked in the corner. each crate of eggs holds 20 eggs. If there is only 1 broken egg in the entire stack of crates, what percent of the crates have broken eggs in them?
Answer: 5%
Step-by-step explanation: If there is one broken egg in each crate (1/20), you would change that to5%
And if there are ten crates, then you see how many eggs there a re total.
(10 × 20 = 200)
If there are 200 eggs and for every 20 eggs there is on broken one, then there will be 10 broken eggs total. or 10/200
convert the fraction to a decimal ( 10 ÷ 200 = .05)
then convert the decimal to a percent. .05 is equal to 5%
PLEAZE RATE BRAINLIEST!!!
3/4 divided by 1/2
multiple choices
2/3
1 1/4
1 1/2
3
please hurry and choose between those
Answer:
Step-by-step explanation:
Diane bought new headphones originally listed for $70.99. They are 25% off. Which equation can be used to find the amount Diane will save?
Step-by-step explanation:
100% = $70.99
there is a discount of 25%.
that means 75% (100 - 25) of the original price remains.
the equation to get any x% amount of a 100% total is simply
x% amount = 100% total amount × x/100
25% = 70.99 × 25/100 = $17.75
Now suppose that not every player can play in every position. The outfielders (left field, center field, right field) can play any outfield position, the infielders (1st base, 2nd base, 3rd base, short stop) can play any infield position, the pitchers can only pitch, and the catchers can only catch. Suppose a certain team has 20 players, of whom 3 are catchers, 4 are outfielders, 6 are infielders, and 7 are pitchers.
How many ways can the team assign field positions to 9 of the 19 players, putting each of the 9 selected players in a position he can play, and ensuring that all 9 field positions are filled?
Answer:
The team can assign field positions to 9 of the 19 players in 181,440 different ways.
Step-by-step explanation:
Since the outfielders (left field, center field, right field) can play any outfield position, the infielders (1st base, 2nd base, 3rd base, short stop) can play any infield position, the pitchers can only pitch, and the catchers can only catch, supposing a certain team has 20 players, of whom 3 are catchers, 4 are outfielders, 6 are infielders, and 7 are pitchers, to determine how many ways can the team assign field positions to 9 of the 19 players, putting each of the 9 selected players in a position he can play, and ensuring that all 9 field positions are filled, the following calculation must be performed:
3 x 7 x 6 x 5 x 4 x 3 x 4 x 3 x 2 = X
21 x 30 x 12 x 24 = X
630 x 12 x 24 = X
181,440 = X
Therefore, the team can assign field positions to 9 of the 19 players in 181,440 different ways.
how to work this problem 736.5 x .78
If angle ACB is congruent, and angle BAC=3x-10, angle ECD=45 and angle EDC=2x+10 what is x?
Answer:
x=20
Step-by-step explanation:
Since BAC and EDC are the same (ACB and DCE are congruent), 2x+10=3x-10, x=20
PLEASE HELP ME AND BE CORRECT BEFORE ANSWERING
TELL ME WHERE TO PUT EACH POINT
Help please :c
What is the probability that a randomly selected student from your group is a 9th grader?
What is the probability that a randomly selected student from your group prefers comedy?
What is the probability that a randomly selected student does not prefer action?
What is the probability that a student prefers comedy, given that the student is in the 10th grade?
What is the probability that a student does not prefer action, given that the student is in 9th grade?
Answer:
1/4
3/8
5/8
2/5
Step-by-step explanation:
From the table :
1.)
P(9th grade) = (9th grade total / total) = 4 / 16 = 1/4
2.)
P(comedy) = (comedy total / total) = 6 / 16 = 3/8
3.)
P(not action) = 1 - P(action) = 1 - (action total / total) = (1 - 6/16) = 1 - 3/8 = 5/8
4.)
P(comefy | 10th grade) = P(comedy n 10th grade) / P(10th grade) = 2 / 5
Answer:
The answer is 12
Step-by-step explanation:
Will give brainliest answer
Answer:
Step-by-step explanation:
The radius of a circular disk is given as 26 cm with a maximum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (Round your answer to two decimal places.) cm2 (b) What is the relative error
Answer:
(a) Hence the maximum error in the calculated area of the disk is 32.67[tex]cm^{2}[/tex].
(b) Hence the relative error is 1.54%.
Step-by-step explanation:
Here the given are,
The Radius of the circle r = 26cm.
The maximum error in measurement dr = 0.2 cm.
Answer:
(a) [tex]A =(4245.28\pm32.66) cm^2[/tex]
(b) [tex]\frac{dA}{A}=\frac{32.66}{4245.28}=0.0077[/tex]
Step-by-step explanation:
radius, r = 26 cm
error = 0.2 cm
(a) The area of the disc is given by
[tex]A = \pi r^2\\\\dA = 2\pi r dr\\\\dA = 2 \times 3.14\times 26\times 0.2= 32.66[/tex]
Now
A = 3.14 x r x r = 3.14 x 26 x 26 = 4245.28 cm^2
So, the area with error is given by
[tex]A =(4245.28\pm32.66) cm^2[/tex]
(b) The relative error is
[tex]\frac{dA}{A}=\frac{32.66}{4245.28}=0.0077[/tex]
What are the x-intercepts of this quadratic function?
g(x) = -2(x - 4)(x + 1)
Answer:
4 and -1
Step-by-step explanation:
A quadratic function is given to us and we need to find the x Intercepts of the given function . The function is ,
[tex]\bf \implies g(x) = -2( x -4)(x+1)[/tex]
Step 1 : Equate the function with 0,
[tex]\bf \implies -2( x -4)(x+1) = 0 [/tex]
Step 2: Equating with 0 :-
Now equate each factors seperately with 0 , to get the x Intercepts.
Step 3: Finding the intercepts :-
[tex]\bf \implies x - 4 = 0 \\\\\implies\boxed{\bf\blue{ x = 4 }} [/tex]
Again ,
[tex]\bf \implies x + 1 = 0 \\\\\implies\boxed{ \blue{\bf x = -1}} [/tex]
Hence the x intercepts are 4 and -1 .
Answer:D (4,0) and (-1,0)
Step-by-step explanation:
Plz help me find sides M and N round to the nearest tenth
Answer:
44
Step-by-step explanation:
gradpoint
Find x please step by step explanation need it
Answer:
37.3
Step-by-step explanation:
19tan(63) = x = 37.289... = 37.3
A student writes
2
pages of a report in
2
an hour. What is her unit
rate in pages per hour?
Answer:
1 page of a report per hour
Step-by-step explanation:
2/2= 1 page per hour
Lets find unit rate
[tex]\\ \sf\longmapsto \dfrac{2\dfrac{1}{2}}{2}[/tex]
[tex]\\ \sf\longmapsto \dfrac{5}{2}\div 2[/tex]
[tex]\\ \sf\longmapsto \dfrac{5}{4}[/tex]
[tex]\\ \sf\longmapsto 1.2pages/hour[/tex]
A line has a slope of -5 and
passes through the point
(0, -7). What is its equation in
slope-intercept form?
Write your answer using integers,
proper fractions, and improper
fractions in simplest form.
Do 8 plz find OS thanks
Answer:
OS = 10
Step-by-step explanation:
We can use a ratio to solve
QP PR
----- = ------
QO OS
14 7
----- = -------
14+6 OS
14 7
----- = -------
20 OS
Using cross products
14 * OS = 7*20
14 OS = 140
Divide by 14
OS = 140/14
OS = 10
In a class there are 400 students. 250 of them studying Mathematics, 150 of them studying English, 100 of them studying Biology, 100 of them studying both Mathematics and English, 60 of them studying both Mathematics and Biology, 40 of them studying both English and Biology, 30 of them studying all Mathematics, English and Biology. Find
a) The number of students who are studying only Mathematics.
b) The number of students who are studying only English.
c) The number of students who are studying only Biology.
d) The number of students who are studying none of Mathematics, English and Biology.
I hope this helps and sorry if it's wrong
Find the total surface area of this pentagon
Answer:
Step-by-step explanation:
2(lb+bh+hl) is the formula for surface area. So,
2(54+81+54) = 2(189) which is 378.
I think the surface area is 378 square inches, or 378 in^2
A kite is a quadrilateral with two pairs of adjacent, congruent sides. The vertex angles are those angles in between the pairs of congruent sides. Prove the diagonal connecting these vertex angles is perpendicular to the diagonal connecting the non-vertex angles. Be sure to create and name the appropriate geometric figures. This figure does not need to be submitted.
a family has two children. what is the probability that both are girls, given that at least one is a boy
Answer: The probability would be zero
Step-by-step explanation:
There are two children and one is a boy, so the probability of both being girls is a zero
A jar of gumballs contains 4 reds, 2 greens, and 6 blues. What is the probability of getting two blues in a row without replacement?
Select one:
a. 3/4
b. 1/2
c. 5/22
d. 5/11
Answer:
C
Step-by-step explanation:
Hypergeometric distribution
[tex]\frac{{6\choose2}}{{12\choose2}}=\frac{15}{66}= \frac{5}{22}[/tex]
ASAP!!!!!! SHOW WORK!!!! Thank you
Answer:
y = -6Step-by-step explanation:
If the three points are collinear, the slopes of RS and ST are same:
m(RS) = (4 - 8)/(1 + 1) = -4/2 = -2m(ST) = (y - 4)/(6 - 1) = (y - 4)/5Since the sloes are equal we have the following equation:
(y - 4)/5 = -2y - 4 = -10y = -10 + 4y = -6Slopes must be equal
slope of RS
m=4-8/1+1m=-4/2m=-2Now
Slope of ST=-2
y-4/6-1=-2y-4/5=-2y-4=-10y=-10+4y=-6The distance, y, in miles, traveled by a car in a certain amount of time, x, in hours, is shown in the graph below:
A graph titled Motion of Car is shown with Time in hours labeled on x-axis and Distance from Starting Point in miles labeled on y-axis. The scale on the x-axis shows the numbers 1, 2, 3, 4, 5, 6, and the scale on the y-axis shows the numbers 0, 14, 28, 42, 56, 70, 84. There are three straight lines in the graph. The first line joins ordered pair 0, 0 with 3, 42. The second straight line joins 3,42 and 4,42 and the third straight line joins ordered pair 4,42 with the ordered pair 5,56.
Which of the following best describes the motion of the car shown?
It travels for 2 hours, then stops for 1 hour, and finally travels again for 5 hours.
It travels for 3 hours, then stops for 1 hour, and finally travels again for 1 hour.
It travels for 3 hours, then stops for 4 hours, and finally travels again for 5 hours.
It travels for 2 hours, then stops for 2 hours, and finally travels again for 1 hour.
Answer:
The last choice
Step-by-step explanation:
:)
The entire graph of the function g is shown in the figure below.
Write the domain and range of g as intervals or unions of intervals.
Step-by-step explanation:
here's the answer to your question
14) The height, h metres, of a ball projected directly upwards from the ground can be modelled by h = 56t - 71, where t is the time in seconds after it leaves the ground. a) Find the height of the ball 3.5 seconds after it leaves the ground. b) At what time will the ball strike the ground again? c) When will the ball be 49 m above the ground? Briefly explain why there are two possible answers.
Using the net below, find the surface area
of the rectangular prism.
5 cm
5 cm
2 cm
5 cm
5 cm
2 cm
2 cm
2 cm
Surface Area =
Answer:
Surface area = 90 cm²
Step-by-step explanation:
Given net of the rectangular prism shows the dimensions as,
Length = 5 cm
Width = 5 cm
Height = 2 cm
`Expression for the surface area of a rectangular prism = 2(lb + bh + hl)
Here, l = length
w = width
h = height
By substituting the values in the expression,
Surface area = 2(5×2 + 5×2 + 5×5)
= 2(10 + 10 + 25)
= 90 cm²
Answer:
Step-by-step explanation:
Describe how to transform the graph of f(x) = x2 to obtain the graph of the related function g(x).
Then draw the graph of g(x).
1. g(x) = f(x + 1)
2. g(x) = f(x) - 2
Please help i also need to graph
9514 1404 393
Answer:
left 1 unitdown 2 unitsStep-by-step explanation:
The transformation g(x) = f(x -h) +k is a translation of f(x) to the right by h units and up k units.
1. h = -1, so the graph of g(x) is the graph of f(x) shifted left 1 unit. (blue)
__
2. k = -2, so the graph of g(x) is the graph of f(x) shifted down 2 units. (green)
Which of the following numbers is rational? Assume that the decimal patterns continue.
9514 1404 393
Answer:
(c) √49
(d) 2.544544...(3-digit repeat)
Step-by-step explanation:
Square roots of perfect squares are rational, as are repeating decimals.
what is the area of a triangle of base 10m and height of 8m
Answer:
40m
Step-by-step explanation:
to find the area of a triangle you must do bh/2
So you do 10 times 8 which is 80.
Then you do 80 divided by 2 which is 40.
I hope this helps!
