Determine the above sequence converges or diverges. If the sequence converges determine its limit
Answers
Answer:
This series is convergent. The partial sums of this series converge to [tex]\displaystyle \frac{2}{3}[/tex].
Step-by-step explanation:
The [tex]n[/tex]th partial sum of a series is the sum of its first [tex]n\!\![/tex] terms. In symbols, if [tex]a_n[/tex] denote the [tex]n\![/tex]th term of the original series, the [tex]\! n[/tex]th partial sum of this series would be:
[tex]\begin{aligned} S_n &= \sum\limits_{k = 1}^{n} a_k \\ &= a_1 + a_2 + \cdots + a_{k}\end{aligned}[/tex].
A series is convergent if the limit of its partial sums, [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex], exists (should be a finite number.)
In this question, the [tex]n[/tex]th term of this original series is:
[tex]\displaystyle a_{n} = \frac{{(-1)}^{n+1}}{{2}^{n}}[/tex].
The first thing to notice is the [tex]{(-1)}^{n+1}[/tex] in the expression for the [tex]n[/tex]th term of this series. Because of this expression, signs of consecutive terms of this series would alternate between positive and negative. This series is considered an alternating series.
One useful property of alternating series is that it would be relatively easy to find out if the series is convergent (in other words, whether [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex] exists.)
If [tex]\lbrace a_n \rbrace[/tex] is an alternating series (signs of consecutive terms alternate,) it would be convergent (that is: the partial sum limit [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex] exists) as long as [tex]\lim\limits_{n \to \infty} |a_{n}| = 0[/tex].
For the alternating series in this question, indeed:
[tex]\begin{aligned}\lim\limits_{n \to \infty} |a_n| &= \lim\limits_{n \to \infty} \left|\frac{{(-1)}^{n+1}}{{2}^{n}}\right| = \lim\limits_{n \to \infty} {\left(\frac{1}{2}\right)}^{n} =0\end{aligned}[/tex].
Therefore, this series is indeed convergent. However, this conclusion doesn't give the exact value of [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex]. The exact value of that limit needs to be found in other ways.
Notice that [tex]\lbrace a_n \rbrace[/tex] is a geometric series with the first term is [tex]a_0 = (-1)[/tex] while the common ratio is [tex]r = (- 1/ 2)[/tex]. Apply the formula for the sum of geometric series to find an expression for [tex]S_n[/tex]:
[tex]\begin{aligned}S_n &= \frac{a_0 \cdot \left(1 - r^{n}\right)}{1 - r} \\ &= \frac{\displaystyle (-1) \cdot \left(1 - {(-1 / 2)}^{n}\right)}{1 - (-1/2)} \\ &= \frac{-1 + {(-1 / 2)}^{n}}{3/2} = -\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\end{aligned}[/tex].
Evaluate the limit [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex]:
[tex]\begin{aligned} \lim\limits_{n \to \infty} S_{n} &= \lim\limits_{n \to \infty} \left(-\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\right) \\ &= -\frac{2}{3} + \frac{2}{3} \cdot \underbrace{\lim\limits_{n \to \infty} \left[{\left(-\frac{1}{2}\right)}^{n} \right] }_{0}= -\frac{2}{3}\end{aligned}}_[/tex].
Therefore, the partial sum of this series converges to [tex]\displaystyle \left(- \frac{2}{3}\right)[/tex].
Related Questions
Can someone please help me?
Answers
dotted line that is at (0,1) then the dotted line goes up to the right and also down to the left so for going up its 1 up 1 over and then also draw a line on the orgin 0,0 that goes in the opposistedirection so left up one to the left one and do that corner to corner that line is filled in and then to get your anwser shade in the area below your secoundline to the bottom of the first line which is going to be bottom left of your graph step-by-step explanation:
A box of cereal weighs 12 ounces and costs $2.16. What is the price of the cereal per ounce?
Answers
Answer:
0.18 cents an ounce
Step-by-step explanation:
Take 2.16 and divide it by 12.
given each pair of similar triangles find the missing value. of LJK=PQR find x
Answers
Answer:
1
Step-by-step explanation:
If s= 6 inches, and the altitude, h, is 21 inches, what is the volume of the pyramid?
Answers
Answer: 42 inches
Step-by-step explanation:
To find the volume you multiply 1/3 of the base which is 6 then with 1/3 is 2 them multiply the altitude which is 21. So 2 times 21 is 42 inches
Answer:
The volume of the pyramid is 42 inches³.
Step-by-step explanation:
The formula is V = ⅓ A H. 6x21=126, 126x1/3=42.
There are 7,500 shares of $10 par value participating 9 percent preferred stock outstanding, and 32,500 shares of common stock outstanding. The common stock cash dividend is $.50 per share, and total dividends of $50,000 are distributed. Remaining dividends are shared equally between common and preferred stockholders. Round your answers to two decimal places, if necessary.
Answers
Answer:
add the answers then subtract
Step-by-step explanation:
hope this is right
Sandy has the numbers 1 to 15 in a box. She randomly chooses one number, and without replacing it, chooses another number from the box. What is the probability that Sandy will choose 2 prime numbers?
A.2/15
B.1/7
C.2/5
D.1/2
Answers
Answer:
A: 2/15
Hope this helped!
Jim has ridden 24 miles of a bike course. The course is 40 miles long. What percentage of the course has Jim ridden so far?
Answers
Answer:
-
Step-by-step explanation:
I’m seriously crushing on you. I’m deeply in love with you girl. Will you give me a chance to prove my love? And you’ll have your best days forever. Happy Valentine’s Day crush girl.
Answers
Answer:
I think you'd be better off simping for sunflowers then girls mate.
Step-by-step explanation:
Lol jk
3(x + 3) − 2 < 4
does anyone know this?
Answers
Answer:3 + ? = 7, 3 + n = 7, 3 + x = 1
Step-by-step explanation:
and so on, where the symbols ?, n, and x represent the number we want to find. We call such shorthand versions of stated problems equations, or symbolic sentences. Equations such as x + 3 = 7 are first-degree equations, since the variable has an exponent of 1. The terms to the left of an equals sign make up the left-hand member of the equation; those to the right make up the right-hand member. Thus, in the equation x + 3 = 7, the left-hand member is x + 3 and the right-hand member is 7.
Answer:
Inequality Form: x < −1
Interval Notation: (−∞, −1)
Step-by-step explanation:
3 (x + 3) −2 < 4
Simplify 3 (x + 3) − 2.
Apply the distributive property.
3x + 3 ⋅ 3 − 2 < 4
Multiply 3 by 3.
3x + 9 − 2 < 4
Subtract 2 from 9.
3x + 7 < 4
Move all terms not containing x to the right side of the inequality.
Subtract 7 from both sides of the inequality.
3x < 4 − 7
Subtract 7 from 4.
3x < −3
Divide each term by 3 and simplify.
x < −1
The result can be shown in multiple forms.
Inequality Form: x < −1
Interval Notation: (−∞, −1)
I will give brainliest to whoever helps, I really need this. pls pls help! thanks!!
Answers
Answer:
[tex]\mathrm{1.}\; m=\mathrm{N/A};\: \mathrm{none}; \: x=14\\\mathrm{2.}\; m=0;\: (0,-5); \:y=-5\\\mathrm{3.}\; m=2;\:(0, -6); \: -2x+y=-6\\[/tex]
Step-by-step explanation:
Problem 1:
The slope of a vertical line is always undefined.The y-intercept of a line occurs when [tex]x=0[/tex]. In this line, [tex]x[/tex] is defined already as [tex]14[/tex] and therefore there is no y-intercept.The equation of a vertical line is [tex]x=n[/tex], where [tex]n[/tex] is some constant.Problem 2:
The slope of a horizontal line is always [tex]0[/tex].The y-intercept of a line occurs when [tex]x=0[/tex]. In this line, [tex]y[/tex] is declared as 5 for [tex]x\in \mathrm{R}[/tex]. Therefore, the y-intercept is [tex](0, -5)[/tex].The equation of a horizontal line is [tex]y=n[/tex], where [tex]n[/tex] is some constant.Problem 3:
The slope of a line given two points can be found using [tex]m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}[/tex]. In this case, the slope is [tex]\frac{11-(-1)}{5-(-1)}=\frac{12}{6}=2[/tex].The y-intercept occurs when [tex]x=0[/tex]. Using the equation we found (refer below), plug in [tex]x=0[/tex] to get the y-coordinate of the y-intercept.In slope-intercept form, the equation of a line is written as [tex]y=mx+b[/tex]. Using [tex]m=2[/tex] and any coordinate this line passes through, we find the equation of this line to be [tex]y=2x-6[/tex]. Rearranging this in standard form ([tex]ax^2+bx+c[/tex]), we get [tex]-2x+y=-6[/tex].Use the following definition of absolute value to prove the given statements: If x is a real number, then the absolute value of x , | x | , is defined by: | x | = { x if x ≥ 0 − x if x < 0 For any real number x , | x | ≥ 0 . Moreover, | x | = 0 implies x = 0 . For any two real numbers x and y , | x | ⋅ | y | = | x y | . For any two real numbers x and y , | x + y | ≤ | x | + | y | .
Answers
Answer:
Proved all parts below.
Step-by-step explanation:
As given ,
|x| = [tex]\left \{ {{x , x\geq 0} \atop {-x, x< 0}} \right.[/tex]
To prove- a) For any real number x , | x | ≥ 0 . Moreover, | x | = 0 ⇒ x = 0
b) For any two real numbers x and y , | x | ⋅ | y | = | x y | .
c) For any two real numbers x and y , | x + y | ≤ | x | + | y | .
Proof -
a)
As given x is a real number
Also , by definition of absolute value of x , we get
| x | ≥ 0
Now,
if |x| = 0
⇒ x = 0 and -x = 0
⇒ x = 0 and x = 0
⇒ x = 0
∴ we get
| x | = 0 ⇒ x = 0
Hence proved.
b)
To prove - | x | ⋅ | y | = | x y |
As we have,
|x| = [tex]\left \{ {{x , x\geq 0} \atop {-x, x< 0}} \right.[/tex]
|y| = [tex]\left \{ {{y , y\geq 0} \atop {-y, y< 0}} \right.[/tex]
|xy| = [tex]\left \{ {{xy , x,y > 0 and x,y < 0} \atop {-xy, x > 0, y< 0 and x <0 , y > 0}} \right.[/tex]
We have 4 cases : i) when x > 0 , y > 0
ii) when x > 0 , y < 0
iii) when x < 0, y > 0
iv) when x < 0, y < 0
For Case I - when x > 0 , y > 0
⇒ |x| = x, |y| = y
⇒|x|.|y| = xy
For Case Ii - when x > 0 , y < 0
⇒ |x| = x, |y| = -y
⇒|x|.|y| = -xy
For Case Iii - when x < 0 , y > 0
⇒ |x| = -x, |y| = y
⇒|x|.|y| = -xy
For Case IV - when x < 0 , y < 0
⇒ |x| = -x, |y| = -y
⇒|x|.|y| = (-x)(-y) = xy
∴ we get , from all 4 cases
| x | ⋅ | y | = | x y |
Hence Proved.
c)
To prove - | x + y | ≤ | x | + | y |
Let
|x| = |x + y - y|
≥ |x + y| - |y| ( Triangle inequality)
⇒ |x| + |y| ≥ |x + y|
Hence Proved.
I NEED ANSWER ASAP!!! is the mapping a fun or not a function
Answers
Answer:
function
Step-by-step explanation:
pls brainly give
The equation of the line
Answers
Answer: y = 2
Step-by-step explanation: Horizontal means y =
Answer:
○ [tex]\displaystyle y = 2[/tex]
Step-by-step explanation:
This is a STRAIT HORISONTAL LINEAR EQUATION, therefore sinse every x-coordinate will have the same y-coordinate of [tex]\displaystyle 2,[/tex] this is the option you would pick.
I am joyous to assist you at any time.
Graph the image of the figure after a dilation with a scale factor of 2 centered at (-7, - 2).
Use the Polygon tool to graph the quadrilateral by connecting all its vertices.
Answers
Answer:
Sorry this is late!
Step-by-step explanation:
The formula shown can be used to find A the amount of money Raul has in his saving account Raul wants to find r the rate of interest his money earns which equation is correctly solved for r
Answers
Answer: r= A-P/ Pt
Step-by-step explanation: ur welcome
The equation [tex]r = \frac{A-P}{pt}[/tex] is correctly solved for r.
Here,
The formula shown can be used to find A the amount of money Raul has in his saving account.
Raul wants to find r the rate of interest his money earns.
We have to find the equation is correctly solved for r.
What is Rate of interest?
The interest rate is the amount a lender charges a borrower and is a percentage of the principal—the amount loaned.
Now,
The formula which shown to find A is;
A = P + Prt
Hence, equation is correctly solved for r is;
[tex]A = P + Prt\\\\A-P = Prt\\\\r = \frac{A-P}{Pt}[/tex]
So, The equation [tex]r = \frac{A-P}{pt}[/tex] is correctly solved for r.
Learn more about the rate of interest visit:
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given the function f(x)=x^3 +x^2-2x+1 what is the resulting function f(x) is shifted to the left one unit
Answers
Answer:
The resulting function is [tex]f(x+1) = x^3 + 4x^2 + 3x + 1[/tex]
Step-by-step explanation:
Suppose we have a function f(x). If we want to shift the function 1 unit to the left, we find f(x+1).
In this question, we have:
[tex]f(x) = x^3 + x^2 - 2x + 1[/tex]
Shifting 1 unit to the left:
[tex]f(x+1) = (x+1)^3 + (x+1)^2 - 2(x+1) + 1[/tex]
[tex]f(x+1) = (x^3 + 3x^2 + 3x + 1) + (x^2 + 2x + 1) - 2x - 2 + 1[/tex]
[tex]f(x+1) = x^3 + (3+1)x^2 + (3+2-2)x + 1 + 1 - 2 + 1[/tex]
[tex]f(x+1) = x^3 + 4x^2 + 3x + 1[/tex]
The resulting function is [tex]f(x+1) = x^3 + 4x^2 + 3x + 1[/tex]
Ms. Davis is doing an activity with her statistics students where she gives them a 202020-question multiple choice test, and they know none of the answers. Students need to guess on every question, and each question has 555 possible choices, 111 of which is correct.
What are the mean and standard deviation of the number of questions that each student gets correct?
You may round your answers to the nearest tenth.
Answers
Answer:
The mean and standard deviation of the number of questions that each student gets correct are 4 and 1.789 respectively (
Answer: mean: 4
standard deviation: 1.79
Step-by-step explanation:
Which expression is equivalent to |b| > 6 ?
b 6
b 6
b > –6 or b –6 and b < 6
Answers
Answer:
number 3. is the correct answer.
Ten little monkeys were jumping on a bed. There is a 35% chance that one will fall off and bump his head. In the bedroom next door, five kangaroos were jumping on a bed. Being more adept at jumping, there is only a 20% that a kangaroo will fall off the bed.
1. What are the chances that a monkey and a kangaroo will fall off the bed?
2. What are the chances that a monkey will not fall off the bed?
3. If the monkeys enjoy this activity every night for an entire week, what are the chances that a monkey falls off the bed every one of the seven nights?
4. What are the chances that if the monkeys jump every day for a week that at least one will fall off of the bed?
5. What are the chances that the kangaroos can get away with jumping on the bed for 4 straight nights until they finally have someone fall off the bed on the night?
6. What are the chances that the kangaroos can jump two nights in a row with no one falling off the bed?
7. The monkeys manage to go a whole week without someone bumping their head. One of the kangaroos insists that they are due for an injury. Another says they must be getting better at their jumping skills. Do you think they're due for a crash?
Answers
Answer:
0.07 ; 0.65 ; 0.00064339296875 ; 3.02 * 10^-12
Step-by-step explanation:
Given :
Number of monkeys, n1 = 10
Probability of fall, off ; p(m) = 35% = 0.35
Number of kangaroos, n2 = 5
Probability of fall, off : p(k) = 20% = 0.2
1.) A monkey and kangaroo falls off;
p(m) * p(k) = 0.35 * 0.2 = 0.07
2.) probability that monkey will not fall off;
1 - p(m) = 1 - 0.35 = 0.65
3.) probability that a monkey falls off every one of seven nights :
P(m) = 0.35 ; for 7 nights ;
0.35 *0.35 * 0.35 * 0.35 * 0.35 * 0.35 * 0.35 = 0.35^7 = 0.00064339296875
4.)
P(x ≥ 1) = p(1) + p(2) +... + p(10)
Using binomial probability calculator :
n = 7*10 = 70 ; p = 0.35
P(x ≥ 1) = 70C1 * 0.35^1 * 0.65^69 = 3.02 * 10^-12
Help me with this please worth 40 points!!!
Answers
Step-by-step explanation:
Since KQ bisects RS, we have KR = 0.5SR.
=> 2x - 8 = 0.5(x - 1)
=> 2x - 8 = 0.5x - 0.5
=> 1.5x = 7.5, x = 5.
Kyle's doctor told him that in order to lose weight, he needs to eat less than 2,000 calories a day. He eats 875 calories for breakfast and than 3 protein smoothies. How many calories can the protein smoothies have to reach his goal?
Answers
Answer:
375 calories each
Step-by-step explanation:
2000-875=1125
1125/3=375
You and a friend go to the local taco joint for lunch. You order three tacos and three burritos and your bill totals $11.25. Your friend's bill is $10.00 for 2 burritos and 4 tacos. Which system of equations can be used to determine the cost of b, a single burrito, and t, a single taco?
A.
3t + 3b = 10
2t + 4b = 11.25
B. 3t+ 3b = 11.25
2t + 4b = 10
C. 3t+ 3b = 10
4t + 2b = 11.25
D. 3t + 3b = 11.25
4t + 2b = 10
Answers
Answer:
d
Step-by-step explanation:
Please help me i only have few hours left i need as much as I can ge. ill get you brainliest and send you free mr.beast merch
Answers
Answer:
26cm²
Step-by-step explanation:
Perimeter
2x - 4 + 5x - 13 + x + 9 = 40
8x - 8 = 40
8x = 40 - 8
8x = 32
x = 4
Sides of the triangle
Height = 2x - 4 = 2 x 4 - 4 = 4cm
Hypotenuse = 5x - 13 = 5 x 4 - 13 = 7cm
Base = x + 9 = 4 + 9 = 13cm
Area of triangle
= 1/2 x base x height
= 1/2 x 13 x 4
= 26cm²
Hope this helps
Answer:
26 cm
Step-by-step explanation:
Consider a company that has two different divisions. The annual profits from the two divisions are independent and have distributions Profit1 ~ N(5, 3^2) and Profit2 ~ N(7, 4^2) respectively. Both the profits are in $ Million. Assume that $1 = Rs. 45
A. Specify a Rupee range (centered on the mean) such that it contains 95% probability for the annual profit of the company.
B. Specify the 5th percentile of profit (in Rupees) for the company
C. Which of the two divisions has a larger probability of making a loss in a given year?
Answers
Answer:
a) Rs 603.68
b) Rs 476.33
c) first division
Step-by-step explanation:
Total profit = profit 1 + profit 2 = P
hence P ~ N(12,74) = [tex]\frac{P-12}{\sqrt{74} } ~ N(0,1)[/tex]
A ) Specifying a Rupee range ( centered on the mean ) that contains 95% probability for annual profit of the company
$13.41 = Rs 603.68
B) specifying the 5th percentile of profit
p = $10.59 ≈ Rs 476.33
C) The division that has a larger probability of making a loss in a given year is the first division
attached below is a detailed solution of the given problem
A): Range containing 95% probability for profit of company is
(Rs. 99M, Rs. 1026M).
B): Rs. 170.1 Million.
C): First division of the company has larger probability of making a loss.
Given that:
$1 = Rs. 45
[tex]Profit_1 \sim N(5, 3^2) \\ Profit_2 \sim N(7, 4^2)[/tex]
Thus,
Company's profit:
[tex]P \sim N( 5+7, 3^2 + 4^2) = N(12, 5^2)[/tex]
A):
95% of the probability lies between 1.96 standard deviations of the mean.
Thus range is:
[tex]= (12 - 1.96\times 5, 12 + 1.96 \times 5)\\\\= (\$2.2M, \$22.8M)\\ \\= (\rm Rs. \: 99M, \rm Rs. \: 1026M)[/tex]
B): Fifth percentile is calculated as:
[tex]P(Z \leq \dfrac{p-12}{5}) = 0.05[/tex]
From p values of z score table, we get:
[tex]\dfrac{p-12}{5} = -1.644\\p = 12 - 8.22 = 3.78\\[/tex]
Thus at $3.78M dollars, or Rs. 170.1M amount, 5th percentile of profit lies.
Or 5th percentile of profit is Rs. 170.1 Million.
C): Loss is when profit < 0
Thus: p < 0
The first division of company, thus have larger probability of making a loss in a given year.
Learn more here:
https://brainly.com/question/13299273
Walmart has soft drinks on sale: 4 packs for $20 Kroger has soft drinks on sale : 3 packs for $18 Which store has the better deal? why?
Answers
Answer:
Walmart has the better deal.
Step-by-step explanation:
If you divide the amount of money and the packs they sell, Walmart is cheaper.
Walmart: 20/4 = 5
Kroger: 18/3 = 6
Answer:
It is Walmart 4 packs for 20$
Step-by-step explanation:
Which expression simplifies to 8x^5?
Answers
Answer:
32768x^5
Step-by-step explanation:
[tex] {8x}^{5} [/tex]
...
The annual amount of rainfall in Ventura County is bounded by 6.3 and 20.5, is continuous and each outcome is equally likely. Use this information to answer the following four questions.
a. Determine P(10.2 < x < 18.7).
b. Determine P(x > 17.1).
c. Determine P(x < 6.3).
d. What is E[x]?
Answers
Answer:
a.0.5984
b.0.2394
c.0
d.13.4
Step-by-step explanation:
We are given that
a=6.3,b=20.5
The pdf of uniform distribution
[tex]f(x)=\frac{1}{b-a}=\frac{1}{20.5-6.3}=0.0704[/tex],6.3<x<20.5
0, otherwise
Now,
a.
P(10.2<x<18.7)=[tex]\int_{10.2}^{18.7}f(x) dx[/tex]
[tex]=0.0704[x]^{18.7}_{10.2}[/tex]
[tex]P(10.2<x<18.7)=0.0704(18.7-10.2)=0.5984[/tex]
b.P(x>17.1)
[tex]P(x>17.1)=\int_{17.1}^{20.5}0.0704dx[/tex]
[tex]=0.0704(20.5-17.1)[/tex]
[tex]=0.2394[/tex]
c.P(x<6.3)=0
d.E[x]=[tex]\frac{a+b}{2}=\frac{6.3+20.5}{2}=13.4[/tex]
I WILL MARK BRAINLIEST!!
Answers
Answer:
i think 108
Step-by-step explanation:
please help me with this
Answers
Answer:
[tex] \frac{17}{8} + \frac{5}{3} = \frac{51 + 40}{24} = \frac{91}{24} [/tex]
hope it can help u~~
??????????????????????
Answers
Answer:
f(8)=91 is the answer
answer is 91