Mathematics Middle School

Does the series converge or diverge? if it converges, what is the sum? SHOW YOUR WORK!!!!!!
equation is in picture.

Does The Series Converge Or Diverge? If It Converges, What Is The Sum? SHOW YOUR WORK!!!!!!equation Is

Answers

Answer 1
Answer:  It converges to   -3

=============================================

Reason:

This is a geometric series with

a = -4 = first termr = -1/3 = common ratio

The template is [tex]a(r)^{n-1}[/tex]

If -1 < r < 1, then the infinite geometric series converges to a finite number. This is because we add on smaller and smaller pieces, which prevents the sum going off to infinity.

In the case of r = -1/3, it fits the interval  -1 < r < 1. In other words -1 < -1/3 < 1 is true.

We'll plug those values into the formula below to wrap things up.

[tex]S = \frac{a}{1-r}\\\\S = \frac{-4}{1-(-1/3)}\\\\S = \frac{-4}{1+1/3}\\\\S = \frac{-4}{3/3+1/3}\\\\[/tex]

[tex]S = \frac{-4}{4/3}\\\\S = -4 \div \frac{4}{3}\\\\S = \frac{-4}{1} \times \frac{3}{4}\\\\S = \frac{-4*3}{1*4}\\\\S = -3\\\\[/tex]

Therefore,

[tex]\displaystyle \sum_{n=1}^{\infty} -4\left(-\frac{1}{3}\right)^{n-1} = -3[/tex]

The final answer is -3.

You can verify the answer by generating partial sums with a spreadsheet. The partial sums should steadily get closer to -3.

Here's a few partial sums.

[tex]\begin{array}{|c|c|c|} \cline{1-3}\text{n} & \text{a}_{\text{n}} & \text{S}_{\text{n}}\\\cline{1-3}1 & -4 & -4\\\cline{1-3}2 & 1.333333 & -2.666667\\\cline{1-3}3 & -0.444444 & -3.111111\\\cline{1-3}4 & 0.148148 & -2.962963\\\cline{1-3}5 & -0.049383 & -3.012346\\\cline{1-3}6 & 0.016461 & -2.995885\\\cline{1-3}7 & -0.005487 & -3.001372\\\cline{1-3}8 & 0.001829 & -2.999543\\\cline{1-3}9 & -0.00061 & -3.000153\\\cline{1-3}10 & 0.000203 & -2.99995\\\cline{1-3}\end{array}[/tex]

The interesting thing is that the partial sums [tex]S_n[/tex]  bounce around -3 while also getting closer to it.

Answer 2
Yes the partial sums should get closer to -3.

Related Questions

help meeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee plsssssssssssssssss

Answers

I can’t read bring it closer on a new post
I can’t really see but i’ll be sure to help you if you could bring it closer !

A submarine hovers at 66 and two-thirds yards below sea level. If it ascends 24 and StartFraction 1 over 8 EndFraction yards and then descends 78 and three-fourths yards, what is the submarine’s new position, in yards, with respect to sea level?

Answers

-2911/24 = -121 7/24
The correct answer is -121 7/24
Give him brainlest he deserves it honestly

And like did you know we’re on a moving rock going through endless space

multiply and simplify is possible (2x+4)(x-2)
SHOW YOUR WORK PLEASE!!!

Answers

Answer:

2x^2-8

Step-by-step explanation:

(2x+4)(x-2)

2x(x-2)+4(x-2

2x^2-4x+4(x-2)

2x^2-4x+4x-8

2x^2-8

2x^2-8

Answer:

2x^2 - 8

Step-by-step explanation:

You have to take one of the equations and distribute it to the other equation. I'll distribute the 1st equation.

(2x+4)(x-2)

2x times x = 2x^2

2x times -2 = -4x

4 times x = 4x

4 times -2 = -8

So you get 2x^2 - 4x + 4x - 8.

- 4x + 4x cancel out each other

So your final answer is 2x^2 - 8

What is the difference of the fractions? Use the number line and equivalent fractions to help find the answer.

Negative 2 and one-half minus (negative 1 and three-fourths)
A number line going from negative 3 to 0 in increments of One-fourth.

Answers

The difference of the given fraction is Negative three-fourths.

Sum and difference of fractions

Given the following fraction difference as shown:

[tex]\sf -2 \dfrac{1}{2} - \huge \text(-1 \dfrac{3}{4} \huge \text)[/tex]

Convert to improper fraction

[tex]\sf -\dfrac{5}{2} + \dfrac{7}{4}[/tex]

Find the LCM

[tex]\sf 2(-5)+\dfrac{7}{4}[/tex]

[tex]\sf -10+\dfrac{7}{4}[/tex]

[tex]\sf -\dfrac{3}{4}[/tex]

Hence, the difference of the given fraction is Negative three-fourths.

Learn more on fraction difference at: https://brainly.com/question/30489902

Select all shapes below that are nets of a pentagonal-based pyramid.

Answers

The answer to this one is D.

Answer:

A and B

Step-by-step explanation:

I took some paper and cut it to the shapes and folded them to see which ones are the answer and got those 2. :3

HELP WILL MARK BRAINIEST
What is the median for the set of data?
71
70
72
74

Answers

Answer:

Step-by-step explanation:

To find the median, we need to arrange the data in ascending order:

70, 71, 72, 74

Since the data set has an even number of values, the median is the average of the two middle values. In this case, the two middle values are 71 and 72. Therefore, the median is:

(71 + 72) / 2 = 143 / 2 = 71.5

So, the median for the given set of data is 71.5.

71.5 is the answer for this question

15 points to whoever gets this right <3

Answers

Answer:

77.5

Step-by-step explanation:

Given:

Scores: 65, 70, 75, 80, 85, 90, 95

Number of Students: 8, 5, 3, 6, 6, 4, 2

Step 1: Calculate the sum

Sum = (65 * 8) + (70 * 5) + (75 * 3) + (80 * 6) + (85 * 6) + (90 * 4) + (95 * 2)

= 520 + 350 + 225 + 480 + 510 + 360 + 190

= 2635

Step 2: Calculate the mean

Mean = Sum / Total Number of Students

= 2635 / (8 + 5 + 3 + 6 + 6 + 4 + 2)

= 2635 / 34

≈ 77.5

Answer:77.5

Step-by-step explanation:

A chemist wants to dilute a 30% phosphoric acid solution to a 15% solution. He needs 10 liters of the 15% solution. How many liters of the 30% solution and water must the chemist use?

Answers

The chemist needs to use 5 liters of the 30% solution and 5 liters of water.

Answer:

30% solution x liters

30%(water) 10-x liters

15% 10 liters

Step-by-step explanation:

HELP ASAP!! A student is painting a brick for his teacher to use as a doorstop in the classroom. He is only painting the front of the brick. The vertices of the face are (−4, 2), (−4, −11), (4, 2), and (4, −11). What is the area, in square inches, of the painted face of the brick?
A. 42 in2
B. 52 in2
C. 104 in2
D. 128 in2

Answers

Answer:

104in2

Step-by-step explanation:

The area, in square inches, of the painted face of the brick is Option C 104 [tex]in^{2}[/tex].

Using the Quadrant system of the Number line

B (-4,2) lies in the 2nd quadrant.

C (-4,-11) lies in the 3rd quadrant

A (4,2) lies in the 1st quadrant

D (4,-11) lies in the 4th quadrant

Length of Side AB= Length of Side CD= Distance between x coordinate of A & B OR Between C & D i.e. 8 in

Length of Side BC= Length of side  AD= Distance between y coordinate of A & D OR Between B & C i.e. 13 in

Area of the Square ABCD= (length of Side AB or CD) X (Length of Side  BC or AD)

                                          = (8 x 13) [tex]in^{2}[/tex]

                                          = 104 [tex]in^{2}[/tex]

Thus, Area of square is 104 [tex]in^{2}[/tex].

To learn more about area: https://brainly.com/question/25292087

A data set is displayed using a dot plot. The range of the data is between 50 and 90, and the distribution is skewed left. Which measure of center, the mean or median, is probably a better representation of the data? Explain your reasoning.

Answers

Apologies for the confusion in my previous response. Given that the distribution is skewed left, the median is likely a better representation of the data's center than the mean.

Skewed left distributions have a longer tail towards the lower values, indicating that there are few extreme values on the left side. These extreme values can have a significant impact on the mean, pulling it towards the left and making it an inaccurate representation of the typical value.

In contrast, the median is less affected by extreme values since it only considers the middle value when the data is sorted. It represents the value that separates the lower 50% of the data from the upper 50%, making it more robust against outliers or extreme values.

Hence, in a skewed left distribution, the median is generally a more reliable measure of the data's center than the mean.

15 points to whoever gets this right:)

Answers

To find the mean you need to add up all of the scores then divide by the number of values in the data set which would give u 77.5 or 77 1/2
77.5 bc of significant figures

Please help, screenshot!

Answers

Answer:

4/7

Step-by-step explanation:

class total = 21

chance of picking one of 12 boys out of 21.

12/21 which can be simpilfied to 4/7

4/7

9 girls + 12 boys = 21
Boys/total = 12/21
12/21 divided by 3 = 4/7
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