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mixas84 [53]
3 years ago
10

Determine determine whether the following geometric series converges or diverges. if the series converges find its sum.

Mathematics
2 answers:
Lilit [14]3 years ago
8 0

For starters,

\dfrac{3^k}{4^{k+2}}=\dfrac{3^k}{4^24^k}=\dfrac1{16}\left(\dfrac34\right)^k

Consider the nth partial sum, denoted by S_n:

S_n=\dfrac1{16}\left(\dfrac34\right)+\dfrac1{16}\left(\dfrac34\right)^2+\dfrac1{16}\left(\dfrac34\right)^3+\cdots+\dfrac1{16}\left(\dfrac34\right)^n

Multiply both sides by \frac34:

\dfrac34S_n=\dfrac1{16}\left(\dfrac34\right)^2+\dfrac1{16}\left(\dfrac34\right)^3+\dfrac1{16}\left(\dfrac34\right)^4+\cdots+\dfrac1{16}\left(\dfrac34\right)^{n+1}

Subtract S_n from this:

\dfrac34S_n-S_n=\dfrac1{16}\left(\dfrac34\right)^{n+1}-\dfrac1{16}\left(\dfrac34\right)

Solve for S_n:

-\dfrac14S_n=\dfrac3{64}\left(\left(\dfrac34\right)^n-1\right)

S_n=\dfrac3{16}\left(1-\left(\dfrac34\right)^n\right)

Now as n\to\infty, the exponential term will converge to 0, since r^n\to0 if 0. This leaves us with

\displaystyle\lim_{n\to\infty}S_n=\lim_{n\to\infty}\sum_{k=1}^n\frac{3^k}{4^{k+2}}=\sum_{k=1}^\infty\frac{3^k}{4^{k+2}}=\frac3{16}

Schach [20]3 years ago
4 0

Answer:

3/16 (converges)

Step-by-step explanation:

Let's write out the first few terms of this sequence, from k=1 to k=3. This gives us:

sum = (3^1)/(4^(1+2)) + (3^2)/(4^(2+2)) + (3^3)/(4^(3+2)) + ...

Computing that into numbers, we have:

sum = 3/64 + 9/256 + 27/1024 + ...

Now, what happens if we multiply both sides by 4/3 (which we get from the 3 and the 4 in the problem)? This gives us:

(4/3)*sum = 4/3*(3/64) + 4/3*(9/256) + 4/3*(27/1024) + ...

which computes out to:

(4/3)*sum = 1/16 + 3/64 + 9/256 + ...

Now, if we subtract sum from 4/3*(sum), notice that most of the terms cancel out. We are left with:

(4/3)*sum - sum = 1/16

Solving this algebraic equation gives us sum = 3/16

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Answer:

x = 3

y = 8

Step-by-step explanation:

Since it gives you x, you need to plug in the number given for it into the equation:

y = 3x - 1

y = 3(3) - 1

y = 9 - 1

y = 8

So now you have, x = 3 and y = 8

5 0
3 years ago
Which equation represents the line that passes through (–6, 7) and (–3, 6)? y = –y equals negative startfraction one-third endfr
egoroff_w [7]

The equation represents the line that passes through (–6, 7) and (–3, 6) \rm y=\dfrac{-1}{3}x+5.

<h3>What is the slope of the equation?</h3>

For all lines in slope y-intercept form, it would be very simple to just find the answer by finding yourself the slope and y-intercept of the line in question.

The slope of the line is;

\rm m = \dfrac{y_2-y_1}{x_2-x_1}\\\\m =\dfrac{7-6}{-6-(-3)}\\\\m = \dfrac{1}{-3}

The equation represents the line that passes through (–6, 7) and (–3, 6) is;

\rm y=\dfrac{-1}{3}x+b\\\\6=\dfrac{-1}{3}(-3)+b\\\\6=1+b\\\\b = 6-1\\\\b=5

The required line of the equation is;

\rm  y =mx+c\\\\y=\dfrac{-1}{3}x+5

Hence, the equation represents the line that passes through (–6, 7) and (–3, 6) \rm y=\dfrac{-1}{3}x+5.

To know more about the equation of line click the link given below.

brainly.com/question/8955867

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Convert 120 degrees to radian measure in terms of pi
Varvara68 [4.7K]
2π radians is equal to 360° which is just π rad/180° so:

120°(π/180°)=2π/3


4 0
3 years ago
A slide 15 feet long makes an angle of 330 with its vertical ladder. To the
Shkiper50 [21]

Answer:

13ft

Step-by-step explanation:

Kindly find attached a rough draft of the situation.

Step one:

Given data

The length of the slide represents the

Hypotenuse of the situation on the rough sketch

Angle =33°

Required

The height of the ladder which is the adjacent of the rough sketch represented by x

Step two:

Applying SOH CAH TOA

Cos θ= adj/hyp

Cos 33=x/15

0.84=x/15

Cross multiplying

x=0.84*15

x=12.6

To the nearest foot the ladder is 13ft tall

6 0
2 years ago
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