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
need help asap please
Answers
Answer:
[tex]\displaystyle 77[/tex]
Step-by-step explanation:
Isosceles triangles will always tell you what you need to know. All angles are congruent, therefore all edges are congruent. ♂️
I am joyous to assist you at any time.
18 percent of what number is 74
Answers
Answer:
411.1111111
Step-by-step explanation:
18% of x = 74
Convert 18% into a decimal.
0.18 • x = 74
Isolate x by dividing by 0.18.
x = 411.1111111
Check answer.
0.18 • 411.1111111 = 74
74 = 74
Please answer the question
Answers
Answer:
D. 2x² + 10xy + 8x² because (2x+ 2y) by (x + 4y)Step-by-step explanation:
Let's mark the sides of the big square as x and a small square as y
See attached picture below
Then we will have the sides of the rectangle:
x + x + y + y = 2x + 2yandx + y + y + y + y = x + 4yThe area of the figure is:
A = lwA = (2x+ 2y)(x + 4y) = 2x² + 8xy + 2xy + 8y² = 2x² + 10xy + 8x²Correct option is D.
Which equation is equivalent to 1/5x - 3 = 2/3
A : x-15=10
B : 3x-15=10
C : 3x-45=10
D : 5x-45=3
Answers
A drag racer has two parachutes, a main and a backup, that are designed to bring the vehicle to a stop (combined with the brakes) after the end of a run. Suppose that the main chute deploys with probability 0 . 95, and that if the main fails to deploy, the backup deploys with probability 0.999.
Required:
a. What is the probability that one of the two parachutes deploys?
b. What is the probability that the backup parachutes deploys?
Answers
Answer:
a. 0.99995 = 99.995% probability that one of the two parachutes deploys
b.0.04995 = 4.995% probability that the backup parachutes deploys
Step-by-step explanation:
a. What is the probability that one of the two parachutes deploys?
Either the first deploys(with probability 95%), or the first does not deploy(probability 5%) and the second does(probability 99.9%). So
[tex]p = 0.95 + 0.05*0.999 = 0.99995[/tex]
0.99995 = 99.995% probability that one of the two parachutes deploys
b. What is the probability that the backup parachutes deploys?
The first does not deploy(probability 5%) and the second does(probability 99.9%). So
[tex]p = 0.05*0.999 = 0.04995[/tex]
0.04995 = 4.995% probability that the backup parachutes deploys
Test your knowledge of lines with the following:
Answers
Ella is Bono motorcycle and he’s taking out a loan in the amount of $15,000 her choice is for the loan or a 36 month loan at 6.50% annual simple interest and 48 month loan at 7.50 annual simple interest what is the difference in the amount of interest Ella would have to pay for these two loans
Answers
Answer:
Step-by-step explanation:
Step-by-step explanation:
The formula for simple interest is expressed as
I = PRT/100
Where
P represents the principal
R represents interest rate
T represents time in years
I = interest after t years
Considering the 36-month loan,
T = 36 months = 36/12 = 3 years
P = $15000
R = 6.5%
Therefore
I = (15000 × 6.5 × 3)/100
I = 292500/100
I = $2925
Considering the 48-month loan loan,
T = 48 months = 48/12 = 4 years
P = $15000
R = 7.5%
Therefore
I = (15000 × 7.5 × 4)/100
I = 450000/100
I = $4500
The difference in the amount of interest Ella would have to pay for these two loans is
4500 - 2925 = $1575
need help asap please
Answers
Step-by-step explanation:
We know that opposite sides of parallelogram are equal so
t + 10 = 3t
3t-t = 10
2t = 10
t = 10/ 2
t = 5
Therefore the value of the should be 5 so that it makes a parallelogram.
Hope it will help :)
I will give brainiest to whoever answers correctly !!
Answers
Answer: approx. 4077.7
Step-by-step explanation:
Use this formula: T=a*(1+m%)^n (T: result; a: initial money; m:interest; n:months)
PLEASE ANSWER ASASP FOR BRAINLEST!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Add six on both sides
-5m >= 30
Divide by negative five on both sides
m >= -6
what is the answer to
2x=24
Answers
Answer:
12
Step-by-step explanation:
you divide 24 by 2 because it is multiplication, and the opposite is division.
Helpppppppppppppppp pleaseeeeee
Answers
290.4 I'm guessing cuz bbn it closest to what I got
Jill orders t-shirts on Amazon. Each shirt costs $15.00 including tax. A one-time shipping fee of $3.25 is added to the cost of the order. The total cost of Jill's order is $78.25. How many t-shirts did Jill order
Answers
Answer:
5 shirts
Step-by-step explanation:
78.25- 3.25 = 75
75/ 15 = 5
Someone please help!
Answers
Answer:
3.25
Step-by-step explanation:
it is shrinking by a factor of 4 so every number is 4 times smaller and 13 divided by 4 will give you 3.25
Answer:
3.25
Step-by-step explanation:
As you can see from the one on the left and the other on the right is that the left is larger than the right by the ratio of 4:1
CB = 12, ZY = 3
BA = 5, YX = 1.25
Using this information, to find ZX, we just need to divide CA's length by 4:
13 / 4 = 3.25
Each number line is shaded to represent a percent drag a percent to each number line that represents a percent.
Answers
Answer:
Step-by-step explanation:
The first one is 20%
The second one is 95%
(Can you take another picture so I can see the rest)
If y=3x+4 and y=2x+8 find the value of both x and y
Answers
Answer:
x=4, y=16
Step-by-step explanation:
Since y is alone on one side of both equations, that means that the equations are equal.
[tex]2x+8=3x+4\\[/tex]
Put like terms on the same side of the equations. I put the numbers on the left and the Xs on the right. To do this subtract 4 from both sides. You will get:
[tex]2x+4=3x[/tex]
Next, I subtracted the 2x from both sides;
[tex]4=x[/tex]
For y, just substitute the x in the equation with 4. Will work for either equation.
[tex]y= 3(4)+4\\y= 12+4\\y=16[/tex]
Or
[tex]y=2x+8\\y=2(4)+8\\y=8+8\\y=16[/tex]
Therefore, y=16
help! asap i’ll mark you brainliest please.
Answers
what is t?
5t - 11 = 99
Answers
Answer:
22
Step-by-step explanation:
5t=99+11
5t=110
divide both side by 5 to find the value of t
A game of chance involves spinning a wheel with 4 number on it. The wheel is designed so that the result of each spin Xhas the following probability distribution. 2 3 Result of a spin .x Probability : 0.50
(a) Find and interpret the mean of X.
(b) Find and interpret the standard deviation of x.
(c) It costs a player $5 for a single spin. The player will receive (in dollars) three times the number that appears. So the profit for one play of this game is Yeur 5. What is the mean and standard deviation of 7
Answers
Distribution table :
X : ___ 1 _____ 2 _____ 3 ______ 4
P(x) __0.50 __0.25 __ 0.15 ____ 0.10
Answer:
1.85 ; 1.014 ;` 0.55 ; 3.042
Step-by-step explanation:
Probability distribution :
X : ___ 1 _____ 2 _____ 3 ______ 4
P(x) __0.50 __0.25 __ 0.15 ____ 0.10
The mean: E(x) = Σ(X * p(x))
(1*0.5) + (2*0.25) + (3*0.15) + (4 *0.10)
= 1.85
Standard deviation = sqrt(Var(x))
Var(x) = Σ(x²*p(x)) - E(x)²
Var(x) = ((1^2*0.5) + (2^2*0.25) + (3^2*0.15) + (4^2 *0.10)) - 1.85^2
= 4.45 - 3.4225
= 1.0275
Standard deviation = sqrt(1.0275)
Standard deviation = 1.0136567
Standard deviation(X) = 1.014
3.)
Cost of spin = $5
Amount, y to be received = 3 times the number that appears
y = 3x - cost of playing
y = 3x - 5
E(y) = E(3x - 5)
E(y) = E(3x) - 5
Recall :E(x) = 1.85
E(y) = 3(1.85) - 5
E(y) = 0.55
Standard deviation :Sd(y) =
Sd(3x - 5)
3(1.014)
= 3.042
Solve for x.
3(-2x-3)+4x-3=8
Answers
Dkdndbxkdkdjxjdbdkdnbx
Hey there!
DISTRIBUTE
3(–2x – 3) + 4x – 3 = 8
3(–2x) + 3(–3) + 4x – 3 = 8
COMBINE LIKE TERMS
–6x – 9 + 4x – 3 = 8
–2x – 12 = 8
ADD by 12 on BOTH SIDES
–2x – 12 + 12 = 8 + 12
CANCEL out: –12 + 12 because that gives 0
KEEP: 8 + 12 because it helps us solve for x
“8+ 12 = 20”
–2x = 20
DIVIDE by –2 on BOTH SIDES
–2x/–2 = 20/–2
CANCEL out: –2/–2 because that gives 1
KEEP: 20/–2 because that gives you x
“x = 20/–2”
Answer: x = –10 ☑️
Good luck on your assignment and enjoy your day!
~LoveYourselfFirst:)
2x ^2 −5=3x
Which of the following describes the solutions to the equation above?
A)The equation has two distinct rational solutions.
B)The equation has two distinct irrational solutions.
C)The equation has one distinct real solution.
D)The equation has no real solutions.
Answers
Answer:a
Step-by-step explanation:
The solutions to the given equation will be two distinct rational solutions, i.e. option A.
What is expression ?Expression is a finite combination of symbols that is well-formed according to rules that depend on the context.
We have,
[tex]2x ^2 -5=3x[/tex]
Now,
Rearrange the equation,
[tex]2x ^2 -3x-5=0[/tex]
Now,
Using the mid term splitting method,
[tex]2x ^2 -3x-5=0[/tex]
i.e.
[tex]2x ^2 -5x+2x-5=0[/tex]
Now,
Taking common,
[tex]2x ^2 +2x-5x-5=0[/tex]
i.e.
[tex]2x (x +1)-5(x+1)=0[/tex]
We get,
[tex](x +1)(2x -5)=0[/tex]
Now,
(x +1) = 0
⇒ x = -1
Now,
(2x -5 )= 0
⇒ [tex]x =\frac{5}{2} =2.5[/tex]
So, the solutions of the given equation are -1 and 2.5.
Hence, we can say that the solutions to the given equation will be two distinct rational solutions, i.e. option A.
To know more about expression click here
https://brainly.com/question/14083225
#SPJ2
Find the sum of whole number that are less than 100 and leave reminder 2 when divided by 5
Answers
Answer:
99
Step-by-step explanation:
2+5, ∈ℤ
=2+7+12+⋯+97
This is AP with =2,=5,
th term is given by =+(−1)
97=2+(−1)5⟹=20
So sum of such integers,
20=202(2+97)=990
The sum of whole number that are less than 100 and leave reminder 2 when divided by 5 is 990.
What is Arithmetic Progression?A series in which the differences between each pair of following terms are the same is known as an arithmetic progression.
An arithmetic progression is a sequence in which every phrase—aside from the initial term—is created by multiplying the previous term by a predetermined amount.
Given:
We have to find the sum of number that are less than 100 and leave reminder 2 when divided by 5
So, we get the sequence
2, 7, 12, ................97
=2+7+12+⋯+97
This is AP with =2,=5,
So, l= a+( n-1)
97= 2+ (n-1)(5)
95 = 5(n-1)
n-1 = 95/5
n-1 = 19
n= 19 + 1
n=20
Now, the sum is
[tex]S_{20}[/tex] = 20/2( 2 x 2+ (20 - 1 )(5))
= 10( 4 + 19(5)
= 10 (99)
= 990
Hence, the sum is 990.
Learn more about Arithmetic Progression here:
https://brainly.com/question/16947807
#SPJ2
Which graph does not represent a function?
Answers
Answer: B
Step-by-step explanation:
It is.
what represents the quantitative forecast
Answers
Answer:
Quantitative forecasting relies on data that can be measured and manipulated. The data is usually from the past. This type of analysis is best for short-term forecasting as making assumptions about the future based on past performance is much more likely to be accurate in the near future.
Step-by-step explanation:
Solve the inequality.
8 + 12x < -16
Answers
Answer:
x < - 2
Step-by-step explanation:
I hope this helps.
Answer:
x < -2
Step-by-step explanation:
1) 12x+8< -16 Rearrange the expression
2) 12x < -16+8 Separate the constants and variables
3) 12x< -24 Add the the number
4) 12x/12 < -24/12 Divide both sides by 12
5) x< -2 Answer
Hope that helps.
Solve for x.
I will give brainliest
Answers
Answer:
22.5
Step-by-step explanation:
Everything equals 180. This means that the 90 degree angle creates half of that would mean that everything on the right side of the 90 degree angle, also has to equal 90 degrees. X=22.5 because you have to combine like terms meaning x + 3x = 90- 4x=90. Divide by 4 on both sides of the equals sign. Plug that into a calculator and you get 22.5 Since it is only asking for the singular x, you don't have to solve for 3x. (It would be 67.5)
can I get an answer please
Answers
Answer:
the second one I'm pretty sure
Step-by-step explanation:
Answer:
its definitely B
Step-by-step explanation:
37
R
12
B
35
С
Find tan(a) in the triangle.
Choose 1 answer:
А
35
37
35
B
12
12
35
12
37
Answers
The value of the tangent of angle α will be 35/12. Then the correct option is B.
What is a right-angle triangle?It's a form of a triangle with one 90-degree angle that follows Pythagoras' theorem and can be solved using the trigonometry function.
The triangle ΔABC is given below.
Then the value of tan α will be
tan α = 35 / 12
Then the correct option is B.
More about the right-angle triangle link is given below.
https://brainly.com/question/3770177
#SPJ1
need help asap please
Answers
Answer:
[tex]\displaystyle yes[/tex]
Step-by-step explanation:
If the angle of elevation is congruent in both triangles, then both smaller legs are congruent.
I am joyous to assist you at any time.
Can someone please explain how to do number 9.
Answers
Answer:
x<15, probably
Step-by-step explanation:
maybe multiply 2 to each side to eliminate the fraction. so 3x+15<60. subtract 15 to isolate the variable term so 3x<45. divide by 3 to isolate x so x<15. check the work. is the inequality true for all values of x less than 15? let's try the first possible integer value, 14, to be sure. 3(14)+15= 57. so 57/2<30. now 57/2 is 28.5 and 28.5<30 so it seems right