Answer:
B. The way in which numbers are grouped to be added does not change the sum.
Answer the question with explanation;
Answer:
The statement in the question is wrong. The series actually diverges.
Step-by-step explanation:
We compute
[tex]\lim_{n\to\infty}\frac{n^2}{(n+1)^2}=\lim_{n\to\infty}\left(\frac{n^2}{n^2+2n+1}\cdot\frac{1/n^2}{1/n^2}\right)=\lim_{n\to\infty}\frac1{1+2/n+1/n^2}=\frac1{1+0+0}=1\ne0[/tex]
Therefore, by the series divergence test, the series [tex]\sum_{n=1}^\infty\frac{n^2}{(n+1)^2}[/tex] diverges.
EDIT: To VectorFundament120, if [tex](x_n)_{n\in\mathbb N}[/tex] is a sequence, both [tex]\lim x_n[/tex] and [tex]\lim_{n\to\infty}x_n[/tex] are common notation for its limit. The former is not wrong but I have switched to the latter if that helps.
Answer:
[tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} = \text{div}[/tex]
General Formulas and Concepts:
Calculus
Limits
Special Limit Rule [Coefficient Power Method]: [tex]\displaystyle \lim_{x \to \pm \infty} \frac{ax^n}{bx^n} = \frac{a}{b}[/tex]Series Convergence Tests
nth Term Test: [tex]\displaystyle \sum^{\infty}_{n = 1} a_n \rightarrow \lim_{n \to \infty} a_n[/tex]Integral Test: [tex]\displaystyle \sum^{\infty}_{n = a} f(n) \rightarrow \int\limits^{\infty}_a {f(x)} \, dx[/tex]P-Series: [tex]\displaystyle \sum^{\infty}_{n = 1} \frac{1}{n^p}[/tex]Direct Comparison Test (DCT)Limit Comparison Test (LCT)Alternating Series Test (AST)Ratio Test: [tex]\displaystyle \sum^{\infty}_{n = 0} a_n \rightarrow \lim_{n \to \infty} \bigg| \frac{a_{n + 1}}{a_n} \bigg|[/tex]Step-by-step explanation:
*Note:
Always apply the nth Term Test as the first test to use for convergence.
Rules:
If [tex]\displaystyle \lim_{n \to \infty} S_n = 0[/tex], then the nth Term Test is inconclusive.If [tex]\displaystyle \lim_{n \to \infty} S_n = l[/tex] (some number l), then the series is divergent by the nth Term Test.Step 1: Define
Identify
[tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2}[/tex]
Step 2: Find Convergence
Substitute in variables [nth Term Test]: [tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} \rightarrow \lim_{n \to \infty} \frac{n^2}{(n + 1)^2}[/tex]Expand: [tex]\displaystyle \lim_{n \to \infty} \frac{n^2}{(n + 1)^2}= \lim_{n \to \infty} \frac{n^2}{n^2 + 2n + 1}[/tex]Evaluate limit [Special Limit Rule - Coefficient Power Method]: [tex]\displaystyle \lim_{n \to \infty} \frac{n^2}{(n + 1)^2} = 1[/tex]Compute [nth Term Test]: [tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} = \text{div}[/tex]∴ by the nth Term Test, the series diverges.
Topic: AP Calculus BC (Calculus I + II)
Unit: Convergence Tests
How far below the surface of the water is the top of a submerged mountain if the ocean floor depth is 12,500 feet and the mountain has a height of 10,190 feet?
Answer:
12500 - 10190 = 2310 ft.
Step-by-step explanation:
Find the linear function f if f^-1(2)=-1 and f^-1 (-9)=3
Answer:
[tex]\displaystyle f(x) = -\frac{11}{4}(x + 1) + 2[/tex]
Step-by-step explanation:
We want to find the linear function given that:
[tex]f^{-1}(2) = -1\text{ and } f^{-1} (-9) = 3[/tex]
Recall that by the definition of inverse functions:
[tex]\displaystyle \text{If } f(a) = b\text{ then } f^{-1}(b) = a[/tex]
In other words, f(-1) = 2 and f(3) = -9.
This yields two points: (-1, 2) and (3, -9).
Find the slope of the linear function:
[tex]\displaystyle m = \frac{\Delta y}{\Delta x} = \frac{(-9) - (2)}{(3) -(-1)} = -\frac{11}{4}[/tex]
From point-slope form:
[tex]\displaystyle y - (2) = -\frac{11}{4}( x- (-1))[/tex]
Hence:
[tex]\displaystyle f(x) = -\frac{11}{4}(x + 1) + 2[/tex]
We can simplify if desired:
[tex]\displaystyle f(x) = -\frac{11}{4}x -\frac{3}{4}[/tex]
solve this equation and show work.
5(c-8)-3(2c+12)=-84
Please help me I nned help
Answer:
length*breadth*height
Which property of equality can be used to justify this step?
15 - 10x = 6x
+ 10x + 10x
————————
15 = 16x
A. Substitution Property of Equality
B. Summation Property of Equality
C. Addition Property of Equality
D. Subtraction Property of Equality
The point A(1,4) has been transformed using the composition T(-1,2) D2. Where is A at?
Answer:
(6, 6)
Step-by-step explanation:
T (-1, 2) = move down 1, right 2
(1, 4) turns into (3, 3)
Dilation scale factor = 2
Multiply x and y value by 2
(3, 3) = (6, 6)
X^2 -24x-108=0 factorize
Answer:
(x-27)(x+4) = 0
Step-by-step explanation:
I think you have a typo, and meant
x^2 -23x-108
what are two factors of 108 that differ by 23?
x^2 -24x-108 does not factor with rational coefficients.
PLEASE HELP NOW
write and solve an equation to find the value of X. please explain in details and show your work!
Answer:
Solution=96°+x+35°
=x+96°+35°
=x=96°-35°
=x=61°
State the gradient of the line 2y = 3 – 2x.
Answer:
Step-by-step explanation:
2y = 3-2x
y = -x + 3/2
Slope of line = coefficient of x = -1
I am still stuck on this basic math question. Rosa has 3 3/4 pounds of dough. She uses 1/8 of a pound for one roll. How many rolls could be made from Rosa's dough? I was told 30. But how did they get this answer?
Step-by-step explanation:
First convert the amount of dough into an improper fraction:
[tex]3\frac{3}{4}\:\text{lbs} = \dfrac{15}{4}\:\text{lbs}[/tex]
Each roll is 1/8 lb so divide the amount of dough by this amount:
[tex]\dfrac{\left(\dfrac{15}{4}\:\text{lbs}\right)}{\left(\frac{1}{8}\:\text{lb/roll}\right)} = \left(\dfrac{15}{4}\:\text{lbs}\right)\cdot \left(8\:\dfrac{\text{roll}}{\text{lb}}\right) = 30\:\text{rolls}[/tex]
Point O is the centroid of triangle ABC. If OD=3x -2 and OC= 5x, find x.
At centroid medians bisect each other in the ratio 2:1
[tex]\\ \sf\longmapsto OC=2(OD)[/tex]
[tex]\\ \sf\longmapsto 2(3x-2)=5x[/tex]
[tex]\\ \sf\longmapsto 6x-4=5x[/tex]
[tex]\\ \sf\longmapsto 6x-5x=4[/tex]
[tex]\\ \sf\longmapsto x=4[/tex]
The value of x is 4, when we are given O is the centroid of the triangle ABC, OD = 3x - 2, and OC = 5x.
What is the centroid of a triangle?The center of the thing is represented by its centroid. The centroid of a triangle is the location where the triangle's three medians connect. The junction of all three medians is another definition for it. The median is a line that connects the middle of a side to the triangle's opposite vertex. The median is divided by the centroid of the triangle in a ratio of 2:1.
How to solve the question?In the question, we are given that O is the centroid of triangle ABC, and are asked to find the value of x, for which OD = 3x - 2, and OC = 5x.
We know that the median is divided by the centroid of the triangle in a ratio of 2:1.
Thus, the median CD is divided by the ratio of 2:1 as,
2/1 = OC/OD,
or, 2/1 = (5x)/(3x - 2),
or, 2(3x - 2) = 1(5x) {Cross-multiplying},
or, 6x - 4 = 5x,
or, 6x - 5x = 4,
or, x = 4.
Thus, the value of x is 4, when we are given O is the centroid of the triangle ABC, OD = 3x - 2, and OC = 5x.
Learn more about the centroid of a triangle at
https://brainly.com/question/8059821
#SPJ2
Find the mean and standard deviation for the set of data. {27, 21, 8, 17, 23,6, 22, 26, 30}
Solve |3n +6| = 21
Answers:
n= -5 and n = 5
n= -9 and n = 9
n= -9 and n = 5
no solution
Answer:
a
Step-by-step explanation:
A cell phone company $500 for a new phone and $60 for a monthly plan. If C(t) is a rational function that represents the average monthly cost of owning the cell phone, what is the range of the function?
A. R: (0,500)
B. R: (60,560)
C. R: R
C: R: (-infinity, infinity)
HOW DO I SOLVE THIS?!
The range of a function is the set of possible values that can be obtained from the dependent variable. The range of the function is: [tex]R: (500, \infty)[/tex]
Given that:
[tex]Phone = \$500[/tex]
[tex]Monthly\ Plan = \$60[/tex]
Let the number of months be t. So, the function C(t) is calculated as follows:
[tex]C(t) = Phone + Monthly\ Plan \times t[/tex]
[tex]C(t) = 500 + 60 \times t[/tex]
[tex]C(t) = 500 + 60t[/tex]
The range is calculated as follows:
The smallest possible value of t is 0 i.e. when no monthly subscription is done.
So, we have:
[tex]C(0) = 500 + 60\times 0= 500 + 0 = 500[/tex]
And the highest is [tex]\infty[/tex] i.e. for a large value of t
So, we have:
[tex]C(\infty) = 500 + 60\times \infty= 500 + \infty = \infty[/tex]
Hence, the range of the function is:
[tex]R: (500, \infty)[/tex]
Read more about range of functions at:
https://brainly.com/question/13824428
Find the LCM of 17, 120 and 240.
-4b+3(6-7b)=93 please help with the asnswr
Answer:
b = -3
Step-by-step explanation:
-4b + 3(6 - 7b) = 93
-4b + 18 - 21b = 93
-4b - 21b = 93 - 18
-25b = 75
[tex]\sf{b = \frac{75}{ - 25} }[/tex]
b = -3
b = - 3
Step-by-step explanation:
-4b + 3(6-7b) = 93
-4b + 18 - 21b = 93
-25b + 18 = 93
-25b = 93 - 18
-25b = 75
b = 75 : - 25
b = - 3
100,000 less than five hundred sixty thousand,three hundred thirteen is __________.
Answer:
100,000>56,313 56, 56,313<100,000
Solve.
6*7-3^2*9+4^3
Every time I did this, I got -103, and my teacher kept saying it's wrong. Please solve and show your work so I can see what I did wrong .
Answer:
25
Step-by-step explanation:
6*7-3^2*9+4^3
Exponents first
6*7-9*9+64
Multiply and divide from left to right
42 -9*9+64
42 - 81 +64
Add and subtract from left to right
-39 +64
25
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\boxed{\mathsf{6\times7-3^2\times9+4^3 }}\\\huge\boxed{\mathsf{= 42 - (9)(9) + 4^3}}\\\huge\boxed{\mathsf{= 42 - 81 + 64}}\\\huge\boxed{\mathsf{= -39 + 64}}\\\huge\boxed{\mathsf{= 25}}}}\\\huge\boxed{\mathsf{Therefore, your\ answer\ is: 25}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
Write the equation of a line that passes through ( 4, 6), and has a slope of 1/3
Answer:
y = 1/3x + 14/3
Step-by-step explanation:
y = mx + b
we know the slope, so the equation would be --> y = 1/3x + b
Now we plug in the coordinates of the given point (4, 6) the line goes through, and solve for b:
y = 1/3x + b
6 = 1/3(4) + b
6 = 4/3 + b
-4/3 -4/3
----------------
14/3 = b
How many sentences does 2 paragraphs have?
Answer ASAP
Answer: depends how much you type
Step-by-step explanation:
John ate 1/6 of a cake. Mike ate 1/12 of it. How much of the cake did the two boys eat?
Answer:
1/4
Step-by-step explanation:
1/6 + 1/12 = 1/4
Answer:
3/3
Step-by-step explanation:
1/6= to 2/12 and 1 /12 = 2 /6
let me see mathematics genius help me answer this question.
Step-by-step explanation: Answer is
(i). 19% or 14/75
(ii) 28% or 7/25
We are asked to find multiple probabilities.
We got to find the number of combinations posible,
Use the combinations formula
[tex]c {}^{n} _r{?} = \frac{(r + n - 1) \: fractorial).}{r \: fractorial(n - 1)fractorial} [/tex]
For math this is read as,
if n choose r,( r+n-1)!/r!(n-1)!.
Where r is how many things we need from and n is the number of things we choose from.
We need 2 things and we have 24 objects to pick from.
So r=2 N equal=24
Which equal
25!/2!(23)!
Which equals
[tex]300[/tex]
So there are 300 possible combinations.
Using
For the 1st question, Since we are given two independent events, we can just multiply the number of good articles by major.
[tex]14 \times 4 = 56[/tex]
So this means the probability is
[tex] \frac{56}{300} = \frac{14}{75} [/tex]
Which is 19%
For the 2nd question, the can multiply the number of minor articles by major articles.
[tex]6 \times 4 = 24[/tex]
So the probability
is
[tex] \frac{24}{300} = \frac{8}{100} = \frac{2}{25} [/tex]
Which is equal to 8%
If R = {(1,2) ,(2,3) ,(3,4) ,(4,5) } is subset of A x B , Find domain (A) and Range (B)
Answer:
hey
here is the attachment of your question
Answer:
[tex]thank \: you[/tex]
When solving the given equation, what is the first step?
(a)
Given: -3 + 2 = 4
А
Add 3 to both sides
B
Subtract 3 from both sides
с
Multiply -3 to both sides
D
Divide -3 from both sides
(b)
What is the solution to the give equation?
Given: -3 + x = 4
x = "write the answer (number only) in the answer box"
Answer:
(a) A
(b) x = 7
Step-by-step explanation:
(a) Given : -3 + 2x = 4
Step 1: Isolate the x to one side. To do so, we get rid of -3 by adding 3 to both sides. (-3 + 3 =0)
2x = 4 + 3
(b) -3 + x = 4
x = 4 + 3
= 7
the set of all natural numbers less than 15 write set build notation
Answer:
{x : x Σ N, x<15}
Step-by-step explanation:
Set of all natural number less than 14 = {x : x Σ N, x<15}
can a math god help me out?
Answer:
[tex]f(1)=70[/tex]
[tex]f(n)=f(n-1)+6[/tex]
Step-by-step explanation:
One is given the following function:
[tex]f(n)=64+6n[/tex]
One is asked to evaluate the function for [tex](f(1))[/tex], substitute [tex](1)[/tex] in place of [tex](n)[/tex], and simplify to evaluate:
[tex]f(1)=64+6(1)[/tex]
[tex]f(1)=64+6[/tex]
[tex]f(1)=70[/tex]
A recursive formula is another method used to represent the formula of a sequence such that each term is expressed as a function of the last term in the sequence. In this case, one is asked to find the recursive formula of an arithmetic sequence: that is, a sequence of numbers where the difference between any two consecutive terms is constant. The following general formula is used to represent the recursive formula of an arithmetic sequence:
[tex]a_n=a_(_n_-_1_)+d[/tex]
Where ([tex]a_n[/tex]) is the evaluator term ([tex]a_(_n_-_1_)[/tex]) represents the term before the evaluator term, and (d) represents the common difference (the result attained from subtracting two consecutive terms). In this case (and in the case for most arithmetic sequences), the common difference can be found in the standard formula of the function. It is the coefficient of the variable (n) or the input variable. Substitute this into the recursive formula, then rewrite the recursive formula such that it suits the needs of the given problem,
[tex]a_n=a_(_n_-_1_)+d[/tex]
[tex]a_n=a_(_n_-_1_)+6[/tex]
[tex]f(n)=f(n-1)+6[/tex]
What method would you choose to solve the
equation 2x2 - 7 = 9? Explain why you chose this
method.
Answer:
» ◻ Add and divide to isolate x²
[tex]2 {x}^{2} - 7 = 9 \\ 2 {x}^{2} = 9 + 7 \\ 2 {x}^{2} = 16 \\ {x}^{2} = \frac{16}{2} \\ \\ {x}^{2} = 8[/tex]
» ◻ use the square root property of equality
[tex] {x}^{2} = 8 \\ x = \sqrt{8} [/tex]
What is the best description of the relationship in the scatterplot below?
Step-by-step explanation:
As x increases. f(x) increases so their is a positive linear association.
Can someone plz solve the math problem i attached as a screenshot. Thx
Answer:
[tex]radius \: of \: first \: circle \: = \frac{3}{2} = 1.5cm \\ radius \: of \: \: second = \frac{4}{2} = 2cm \\ radius \: of \: third = \frac{6}{2} = 3cm \\ then \: ab \: length \\ = radius \: of \: first + diameter \: of \: second \: +r adius \: of \: third \\ = 1.5 + 4 + 3 \\ = 1.5 + 7 \\ = 8.5 cm\\ thank \: you[/tex]