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
LiRa [457]
4 years ago
9

Write out the first four terms of the series to show how the series starts. Then find the sum of the series or show that it dive

rges. Summation from n equals 0 to infinity (StartFraction 9 Over 7 Superscript n EndFraction plus StartFraction 3 Over 5 Superscript n EndFraction )
Mathematics
1 answer:
Nostrana [21]4 years ago
4 0

Answer:

The first four terms of the series are

(9+3),(\frac97+\frac35),(\frac9{7^2}+\frac3{5^2}),(\frac9{7^3}+\frac3{5^3})

\sum_{n=0}^\infty \frac9{7^n}+\frac{3}{5^n} = 14.25

Step-by-step explanation:

We know that

Sum of convergent series is also a convergent series.

We know that,

\sum_{k=0}^\infty a(r)^k

If the common ratio of a sequence |r| <1 then it is a convergent series.

The sum of the series is \sum_{k=0}^\infty a(r)^k=\frac{a}{1-r}

Given series,

\sum_{n=0}^\infty \frac9{7^n}+\frac{3}{5^n}

=(9+3)+(\frac97+\frac35)+(\frac9{7^2}+\frac3{5^2})+(\frac9{7^3}+\frac3{5^3})+.......

The first four terms of the series are

(9+3),(\frac97+\frac35),(\frac9{7^2}+\frac3{5^2}),(\frac9{7^3}+\frac3{5^3})

Let

S_n=\sum_{n=0}^\infty \frac{9}{7^n}    and     t_n=\sum_{n=0}^\infty \frac{3}{5^n}

Now for S_n,

S_n=9+\frac97+\frac{9}{7^2}+\frac9{7^3}+.......

    =\sum_{n=0}^\infty9(\frac 17)^n

It is a geometric series.

The common ratio of S_n is \frac17

The sum of the series

S_n=\sum_{n=0}^\infty \frac{9}{7^n}

    =\frac{9}{1-\frac17}

    =\frac{9}{\frac67}

    =\frac{9\times 7}{6}

    =10.5

Now for t_n

t_n= 3+\frac35+\frac{3}{5^2}+\frac3{5^3}+.......

    =\sum_{n=0}^\infty3(\frac 15)^n

It is a geometric series.

The common ratio of t_n is \frac15

The sum of the series

t_n=\sum_{n=0}^\infty \frac{3}{5^n}

    =\frac{3}{1-\frac15}

    =\frac{3}{\frac45}

    =\frac{3\times 5}{4}

    =3.75

The sum of the series is \sum_{n=0}^\infty \frac9{7^n}+\frac{3}{5^n}

                                        = S_n+t_n

                                       =10.5+3.75

                                       =14.25

You might be interested in
What is the negative square root of 64/4?
Goryan [66]

4/64

then that simplifies to 1/16

8 0
3 years ago
Parallelogram ABCD is shown.<br> CD=DE<br> Work out the size of angle x.
Andru [333]

Step-by-step explanation:

Angles in a parallelogram are supplementary

7 0
3 years ago
Read 2 more answers
Order each set of values from least to greatest 0.4 and 5/8 and 38%
Zarrin [17]
38%, 0.4, 5/8.
Would you like me to explain
6 0
3 years ago
Read 2 more answers
The circumference is 307.72 yards. What is the diameter?
aivan3 [116]

I think its ans is

30 inches

6 0
3 years ago
Read 2 more answers
Which of the given is irretional number? OPTIONS:A.)√256 B.)√225/576 C.)√3 D.)√9×16 .​
RideAnS [48]

Answer:

no

Step-by-step explanation:

3 0
3 years ago
Other questions:
  • Use part 1 of the fundamental theorem of calculus to find the derivative of the function. g(x) = x e2t2 − 4t dt 4
    12·1 answer
  • Which is greater 0.75 or 0.9
    11·2 answers
  • Can someone please help me with this?
    15·2 answers
  • A square game board is divided into smaller squares, each with sides one-third the length of the sides of the board.
    9·1 answer
  • Below is the graph of a polynomial. Which of the following statements about<br> this graph is true?
    6·1 answer
  • Evaluate the following expression x^2+x when x=5
    12·1 answer
  • Ty had 50 tickets for games at a carnival.He used 1/5 of the tickets to play the ball-toss game.He then used 1/2 of the remainin
    11·2 answers
  • Hi help me please ,,
    11·2 answers
  • What is this number in standard form? seventy-one and ninety-two thousandths
    6·2 answers
  • PLEASE HELP WILL GIVE BRIANLIEST!!! ​
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!