1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Dominik [7]
2 years ago
5

Identify whether the series summation of 8 open parentheses 5 over 6 close parentheses to the i minus 1 power from 1 to infinity

is a convergent or divergent geometric series and find the sum, if possible. (2 points)
Mathematics
2 answers:
musickatia [10]2 years ago
5 0
<h2>Answer:</h2>

Yes, the series is convergent.

The sum is: 48

<h2>Step-by-step explanation:</h2>

The series is given by:

\sum_{i=1}^{\infty} 8(\dfrac{5}{6})^{i-1}

Now, each term of the series is a constant multiple of the preceding element of the series.

The constant multiple is: 5/6<1

( Since,

a_1=8(\dfrac{5}{6})^{1-1}\\\\i.e.\\\\a_1=8(\dfrac{5}{6})^0\\\\i.e.\\\\a_1=8

a_2=8(\dfrac{5}{6})^{2-1}\\\\i.e.\\\\a_2=8(\dfrac{5}{6})\\\\i.e.\\\\a_2=\dfrac{5}{6}a_1

Similarly, for nth term we have:

a_n=\dfrac{5}{6}a_{n-1}

)

Hence, the series is a geometric series.

Also, this series is convergent.

( since the constant multiple i.e. the common ratio is less than 1)

We know that the sum of the infinite geometric series of the type:

\sum_{n=1}^{\infty}ar^{n-1}=\dfrac{a}{1-r}

where a is the first term of the series and r is the common ratio.

Here we have:

a=8\ and\ r=\dfrac{5}{6}

Hence, we have:

\sum_{i=1}^{\infty} 8(\dfrac{5}{6})^{i-1}=\dfrac{8}{1-\dfrac{5}{6}}\\\\i.e.\\\\\sum_{i=1}^{\infty} 8(\dfrac{5}{6})^{i-1}=\dfrac{8}{\dfrac{6-5}{6}}\\\\i.e.\\\\\sum_{i=1}^{\infty} 8(\dfrac{5}{6})^{i-1}=\dfrac{8}{\dfrac{1}{6}}\\\\i.e.\\\\\sum_{i=1}^{\infty} 8(\dfrac{5}{6})^{i-1}=8\times 6\\\\i.e.\\\\\sum_{i=1}^{\infty} 8(\dfrac{5}{6})^{i-1}=48      

Leto [7]2 years ago
3 0
\displaystyle\sum_{i=1}^\infty\8\left(\dfrac56\right)^{i-1}

This is a geometric series with a common ratio between successive terms that is less than 1 in absolute value. This means the series will converge. The value of the sum is \dfrac8{1-\frac56}=48.

You might be interested in
Subjects for the next presidential election poll are contacted using telephone numbers in which the last four digits are randoml
sammy [17]

Answer:

probability at least one zero is 0.3439

Step-by-step explanation:

given data

last four digits = randomly selected​

to find out

probability that for one such phone​ number the last four digits include at least one 0.

solution

we know there are total 10 digit

so we first find probability of non zero digit i.e.

Probability  ( non zero ) = 9 /10 = 0.9

and now we find probability of none of digit zero  only event happen n= 4 time in a row by multiplication rules i.e

Probability ( none zero in 4 digit ) = 0.9^{n}

Probability ( none zero in 4 digit ) = 0.9^{4}

Probability ( none zero in 4 digit ) = 0.6561

so we can say probability at least one zero = 1 - Probability ( none zero in 4 digit )  

probability at least one zero = 1 - 0.6561

probability at least one zero is 0.3439

3 0
3 years ago
The sun and volcanic eruptions influence earth's climate . choose one of these influences and research the effects it has on ear
sladkih [1.3K]
Volcanic eruptions release molten rock or lava that can form new rock on the earth's surface. The gases,dust, and ashes that erupt from a volcano also affect the climate by covering the earth's surface; therefore, creating a cooling effect on the location of the volcano. Also, in the past where there were super volcanic eruptions that it caused global warming due to volcanoes also releasing greenhouse gases.
8 0
2 years ago
Evaluate 5.5r8.35s when r12 and s4.
Over [174]

Answer:

32.6

Step-by-step explanation:

when you plug in for s and r you will get 66-33.4 since 5.5 multiped by 12 is 66 and 8.35 multiplied by 4 is 33.4 .

then you subtract them and that's how you'll get your answer

3 0
2 years ago
A test has 40 questions. Denise finished 2\5 of the questions in the first 10 minutes and 1\2 of the total questions in the next
leonid [27]

Answer:

28 questions answered.

Step-by-step explanation:

40(2/5) is 16 questions.

40-16 is 24 questions left.

24(.5) is 12 questions.

16+12 gets you 28 questions.

8 0
2 years ago
Estimate the amount of gas left in this cars gas tank with a fraction
Rufina [12.5K]
It looks a little less than 1/2 or 4/8, so probably 3/8 of the gas is left.
3 0
3 years ago
Other questions:
  • Two types of plastics are suitable for an electronics component manufacturer to use. The breaking strength of this plastic is im
    13·1 answer
  • Dwayne earns $11.45 per hour.Last week he worked 38 1/4 hours.how much did he earn last week?
    13·2 answers
  • Please help me <br> Show your work
    8·2 answers
  • Help Please.......Seriously
    7·1 answer
  • Trent ordered sandwiches for his friends. The veggie sandwiches cost $4 apiece and the meat sandwiches cost $5 apiece. Trent ord
    14·1 answer
  • The function g(x) = x2 – 10x + 24 is graphed on a coordinate plane. Where will the function cross the x-axis?
    5·1 answer
  • How do you find LCM? please give a good explanation!
    6·1 answer
  • Find the minimum or maximum value of the function g(x)=−3x2−6x+5
    8·1 answer
  • HELP!! I know how to find the answer but I don't know how to use the number line to find the answer.
    12·1 answer
  • Alison had lost some puzzle pieces. She found 3/10 of them under the coffee table and another 4/10 of them under the couch. How
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!