Determine the above sequence converges or diverges. If the sequence converges determine its limit
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].
Two cars depart an intersection.
One goes north.
The other goes west.
Each car travels at a constant speed of 30 miles per hour.
Assuming the roads go on forever, what will be the distance between the cars after 1 hours and 30 minutes in miles?
Answer:
The distance between them after 1 hour and 30 minutes will be of 63.64 miles.
Step-by-step explanation:
This question can be solved using the Pythagorean Theorem.
Both cars travel at a constant speed of 30 miles per hour. So, after 1 hour and 30 minutes, which is 1 and a half hour, they will have traveled 30*1.5 = 45 miles.
The distance between them is the hypotenuse of a right triangle, in which the sides are 45. So
[tex]d^2 = 45^2 + 45^2[/tex]
[tex]d = \sqrt{45^2 + 45^2}[/tex]
[tex]d = 63.64[/tex]
The distance between them after 1 hour and 30 minutes will be of 63.64 miles.
A person invested $30,000 in stocks and bonds. Her investment in bonds is $3000 more than half her investment in stocks
How much did she invest in stocks? How much did she invest in bonds?
Answer:
He invested in stocks Say, = x $
He invested in bonds = (x/2)+6000
So,
x+ x/2 +6000=30000
2x+x+12000=60000
3x= 48000
x=16000
Investment in stocks= 16000 $
Step-by-step explanation:
!!
Answer:
$ 3 0 , 0 0 0 in Stocks
$ 2 7 , 0 00 in Bonds
Step-by-step explanation:
$ 3 0 , 0 0 0
- $ 3 , 0 0 0
_ _ _ _ _ _ _ _ _ _
$ 2 7 , 0 0 0
cuanto es 2/15 ÷ 4/5=
Answer:
2/15 * 5/4= 10/60 simplify 1/6
Step-by-step explanation:
:)
Answer:
.2
Step-by-step explanation:
What is the value of (4)5/(4)3?
The equation of line t is y = -2x + 9. Perpendicular to line t is line u, which passes through
the point (-10, 4). What is the equation of line u?
Write the equation in slope-intercept form. Write the numbers in the equation as proper
fractions, improper fractions, or integers.
Answer:
Equation of line u in slope-intercept form is: [tex]\mathbf{y=\frac{1}{2}x+9 }[/tex]
Step-by-step explanation:
Equation of line t : y = -2x + 9.
We need to find equation of line u, which is perpendicular to line t and passes through point (-10,4)
The equation must be in slope-intercept form.
The general equation of slope-intercept form is: [tex]y=mx+b[/tex] where m is slope and b is y-intercept
Finding Slope:
If two lines are perpendicular, their slopes are opposite i.e [tex]m=-\frac{1}{m}[/tex]
Slope of line t: y=-2x+9 we get m =-2 (Comparing with general form y=mx+b, we get m =-2)
Slope of line u: [tex]\frac{1}{2}[/tex]
So, we get Slope of line u: m= [tex]\frac{1}{2}[/tex]
Finding y-intercept:
Using slope m= [tex]\frac{1}{2}[/tex] and point(-10,4) we can find y-intercept
[tex]y=mx+b\\4=\frac{1}{2}(-10)+b\\4=-5+b\\b=4+5\\b=9[/tex]
Equation of line u:
So, equation of line u, having slope m= [tex]\frac{1}{2}[/tex] and y-intercept b=9, we get:
[tex]y=mx+b\\y=\frac{1}{2}x+9[/tex]
So, Equation of line u in slope-intercept form is: [tex]\mathbf{y=\frac{1}{2}x+9 }[/tex]
Someone Please help solve this asap
Answer:
Part A: Peter is, based on the graph, Peter's line is steeper(or has a higher slope) than Gina's.
Part B: Gina, She started with 70, while Peter started with 20. Based on the graph
Part C: They had the same amount of money on week 10, both of their dots are on the same point. Based on the graph.
Part D: Y = 10x + 20.
Part E: Y = 5x + 70
Step-by-step explanation:
A teacher is 5 feet 8 inches tall and she shrinks to 9 inches. How tall will her pencil be if it undergoes the same scale shrink and was originally 12 inches?
A) 1.59 inches
B) 1.86 inches
C) 2.68 inches
D) 6.8 inches
Answer:
c
Step-by-step explanation:
2.68
12. Which of the relations below is a function?
A. {(1, 1,), (2, 1), (3, 1), (4, 1), (5, 1)}
B. {(3, 1), (2, 2), (3, 3), (2, 4), (2,5)}
C. {(2, 1), (2, 2), (2, 3), (2, 4), (2,5)}
D. {(1, 1), (2, 2), (2, 3), (2, 4), (1,5)}
Answer:
A
Step-by-step explanation:
a function may not have the same values of X.
Need help with this
GIVE ME THE CORRECT ANSWER I THANK, GIVE 5 STARS, AND GIVE BRAINLIEST!!!!
System of equations
Solve by substitution:
y = x + 7
2x + y = -5
Group of answer choices
(4 , -3)
(4 , 3)
(-4 , 3)
(-4, -3)
Answer:
C: (-4,3)
Step-by-step explanation:
2x+x+7=-5
3x+7=-5
3x=-12
x=-4
y=-4+7
y=3
(-4,3)
solve for (0)
12
15
Round to the nearest hundredth
Answer:
.014
Just trust me, you know I have your back
A manufacturing company has 1000 employees. 10% of the employees earn less than
Rs.500 per day, 200 earn between Rs.500 and Rs.999; 30% earn between Rs.1,000
and Rs.1,499; 250 employees earn between Rs.1,500 and 1,999 and rest earn Rs.2,000
and above. Calculate the suitable average wage. Give the reason for your choice of
average.
Answer:
heehhe
Step-by-step explanation:
vwhehehhehdhdhd
PLEASE HURRY I'LL MARK BRAINLIST!!!!!!!!!!!!!!!!!!!!!!!!!!!!1 Bo, Mike, and Cat are playing a game involving decimals in math class. So far, they have scored 100.2 points combined. Bo scored 38.65 points and Mike scored 29 points. How many points did Cat score?
A fraction bar labeled C divided into 3 parts: Bo, B, and Cat. Bo has A points. B has 29 points. Cat has question mark point.
How many points did Bo score, as represented by A?
Who scored 29 points, as represented by B?
How many points did the 3 players score all together, as represented by C?
What will the answer to the question be?
Answer:
67.65
Step-by-step explanation:
Answer
cat scored: 38.65+29=67.65
100.2-67.65=32.55#
bo 29 marks
Step-by-step explanation:
plz help will make brainliest!!!
Answer:
I think it is B.
2(3−8y)= i need help someone
Answer:
6-16y
Step-by-step explanation:
Distribute
2*3
2*-8y
6-16y
Answer:
this is the answer 2 • (3 - 8y)
Step-by-step explanation:
please help i am so confused
fill in the blanks _ + _ = 10.5
Answer:
there can be many answerrrs I guess
.
Step-by-step explanation:
5.25+5.25=10.5
2.5+8=10.5
have a great day :)
An art class is making a mural for their school which has a triangle drawn in the middle. The length of the bottom of the triangle is x. Another side is 13 more than two times the length of the bottom of the triangle. The last side is 11 more than the bottom of the triangle. Write and simplify an expression for the perimeter of the triangle.
Answer:
P = 4x + 24Step-by-step explanation:
Sides of the triangle:
bottom = xside1 = 2x + 13side2 = x + 11Perimeter is the sum of side lengths:
P = x + 2x + 13 + x + 11 = 4x + 24What is the result when the number 52 is decreased by 50%?
Answer: You get 26, If this wasn't what you were looking for let me know! :)
Step-by-step explanation:
Someone help me??? Would be nice
I believe it's A. B. and C.
Step-by-step explanation:
B is closest to y-axis
A isn't aligned with b
Find the missing angle.
Answer:
[tex]m\angle XYV=135^{\circ}[/tex]
Step-by-step explanation:
Notice that [tex]m\angle XYV=m\angle XYW+m\angle VYW[/tex].
The diagram already marks [tex]m\angle VYW=80^{\circ}[/tex], so we just need to find [tex]m\angle XYW[/tex].
[tex]\angle TYU[/tex] and [tex]\angle XYW[/tex] are vertical angles, which means they are equal in measure.
Therefore,
[tex]m\angle TYU=m\angle XYW=6x+7[/tex]
Since there are [tex]180^{\circ}[/tex] on each side of a line, we have the following equation:
[tex]m\angle XYW+m\angle VYW+m\angle UYV=180^{\circ},\\6x+7+13+4x+80=180,\\10x+20=100,\\10x=80,\\x=8[/tex].
Plugging in [tex]x=8[/tex], we have:
[tex]m\angle XYW=6(8)+7=55^{\circ}[/tex].
Therefore,
[tex]m\angle XYV=55^{\circ}+80^{\circ},\\m\angle XYV=\fbox{$135^{\circ}$}[/tex].
Mel earns $15 per hour working at a bakery. How much does he earn if he
works for 5 hours?
A. $30
B. $155
C. $75
O D. $105
Cuanto 5 entre 6'658
Solve for p. U=1/7pd
Answer:
U =1/7pd
Multiplying through by 7pd, we have U7pd=1
Dividing through by U7d , we have p=U7d*1
Therefore p= U7d
help me.... this is do in 13 min
Answer:
d
Step-by-step explanation:
Find the greatest common factor of these two expressions :
20u^8 y^7 x^6 and 12u^3 y^4
1.Interest rate=7%p.a.Find the interest of Rs 200 for years.
2.Interest rate=11%p.a.Find the interest of Rs 2000 for years.
Step-by-step explanation:
Question does not specify how many year and is in plural form considering the level of question I will consider it is some typo error and is just 1 year
Step 1
Finding Intrest for Rs200
We have
Principal(P) = Rs 200
Rate (R) = 7%pa
Time (T) = 1year
Intrest(I) =?
We have ,
I =PTR/100
=200*7/100
=Rs 14
Step 2
Finding intrest for Rs2000
Given
Principal(P)=Rs2000
Rate(R) =11%
Time(T) =1 year
Intrest(I) =?
We know,
I =PTR/100
=2000*11/100
= 20*11
=Rs 220
PLEASEEEEEE HELPPPPPPP
What types of jobs were Deaf people frequently hired for during the wars?
health care
part time
construction
factory
Answer:
Factory
Step-by-step explanation:
Answer:
factory
Step-by-step explanation:
Identify the coefficient in the algebraic expression 4ac-3
4
-3
3
4ac
Answer:
4
Step-by-step explanation:
4 is the coefficient because its infront of the variable