Answers
Answer:
The statement in the question is wrong. The series actually diverges.
Step-by-step explanation:
We compute
[tex]\lim_{n\to\infty}\frac{n^2}{(n+1)^2}=\lim_{n\to\infty}\left(\frac{n^2}{n^2+2n+1}\cdot\frac{1/n^2}{1/n^2}\right)=\lim_{n\to\infty}\frac1{1+2/n+1/n^2}=\frac1{1+0+0}=1\ne0[/tex]
Therefore, by the series divergence test, the series [tex]\sum_{n=1}^\infty\frac{n^2}{(n+1)^2}[/tex] diverges.
EDIT: To VectorFundament120, if [tex](x_n)_{n\in\mathbb N}[/tex] is a sequence, both [tex]\lim x_n[/tex] and [tex]\lim_{n\to\infty}x_n[/tex] are common notation for its limit. The former is not wrong but I have switched to the latter if that helps.
Answer:
[tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} = \text{div}[/tex]
General Formulas and Concepts:
Calculus
Limits
Special Limit Rule [Coefficient Power Method]: [tex]\displaystyle \lim_{x \to \pm \infty} \frac{ax^n}{bx^n} = \frac{a}{b}[/tex]Series Convergence Tests
nth Term Test: [tex]\displaystyle \sum^{\infty}_{n = 1} a_n \rightarrow \lim_{n \to \infty} a_n[/tex]Integral Test: [tex]\displaystyle \sum^{\infty}_{n = a} f(n) \rightarrow \int\limits^{\infty}_a {f(x)} \, dx[/tex]P-Series: [tex]\displaystyle \sum^{\infty}_{n = 1} \frac{1}{n^p}[/tex]Direct Comparison Test (DCT)Limit Comparison Test (LCT)Alternating Series Test (AST)Ratio Test: [tex]\displaystyle \sum^{\infty}_{n = 0} a_n \rightarrow \lim_{n \to \infty} \bigg| \frac{a_{n + 1}}{a_n} \bigg|[/tex]Step-by-step explanation:
*Note:
Always apply the nth Term Test as the first test to use for convergence.
Rules:
If [tex]\displaystyle \lim_{n \to \infty} S_n = 0[/tex], then the nth Term Test is inconclusive.If [tex]\displaystyle \lim_{n \to \infty} S_n = l[/tex] (some number l), then the series is divergent by the nth Term Test.Step 1: Define
Identify
[tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2}[/tex]
Step 2: Find Convergence
Substitute in variables [nth Term Test]: [tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} \rightarrow \lim_{n \to \infty} \frac{n^2}{(n + 1)^2}[/tex]Expand: [tex]\displaystyle \lim_{n \to \infty} \frac{n^2}{(n + 1)^2}= \lim_{n \to \infty} \frac{n^2}{n^2 + 2n + 1}[/tex]Evaluate limit [Special Limit Rule - Coefficient Power Method]: [tex]\displaystyle \lim_{n \to \infty} \frac{n^2}{(n + 1)^2} = 1[/tex]Compute [nth Term Test]: [tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} = \text{div}[/tex]∴ by the nth Term Test, the series diverges.
Topic: AP Calculus BC (Calculus I + II)
Unit: Convergence Tests
Related Questions
a meayerologist at television station reported that a town received 0.90 inches of rain. Which fraction is equivalent to this amount of rain in inches?
Answers
Answer:
9/10
Step-by-step explanation:
0.9 = 90/100 9/10
0.90 is the same as 90/100
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
A property sells for $140,000 two years after it was purchased. If the annual appreciation rate is 6%, how much did the original buyer pay for it?
Answers
Answer:
$123,200
Step-by-step explanation:
0.06 x 2 = 0.12
140,000 x 0.12 = 16,800
140,000 - 16,800 = 123,200
The original buyer paid the amount of $123,200 for the property.
What is the percentage?The percentage is defined as a ratio expressed as a fraction of 100.
The property appreciated by 6% per year, so in two years it would have appreciated by 6% × 2 years = 12%.
Once we know the total appreciation, we can subtract that value from 100% to find the original purchase price as a percentage of the final sale price.
If the property appreciated by 12%, then the original purchase price must have been 100% - 12% = 88% of the final sale price.
Now, multiply the final sale price by the original purchase price as a percentage to find the original purchase price.
So, the original buyer paid the amount = 88% × $140,000 = $123,200
Therefore, the original purchase price was $123,200.
Learn more about the percentages here:
brainly.com/question/24159063
#SPJ2
For what value(s) of x is f(x)=3?
Answers
Answer:
D
Step-by-step explanation:
Remember that for this kind of question, when f(x) is given as 3, then 3 is the y value of the point. So you must have a point whose value is (x,3)
There is only one point that does that. x = - 2
The answer is D
Two cars start at the same point. One travels north at 65 km/h and the other travels east at 50 km/h. How far are they apart after 30 minutes?
Answers
Answer:
they travel 115 km/h long
The diameter of a circle is 19 7/8". What is the circumference (to 2 decimal places)?
Please note:
C = πd
Answers
Answer:
62.41
Step-by-step explanation:
19 7/8= 19.875
C= pie x d
C= 3.14 x 19.875
C= 62.4075
C= 62.41 (if you want 2 decimal places)
1. If x = y and x + 2y = 6 then find value of 'y'.
T
Answers
Step-by-step explanation:
x=y
x+2y=6
y+2y=6
3y=6
y=2
Step-by-step explanation:
x=y
x+2y=6
y+2y=6
3y=6
y=2
Slope-Intercept Form
Directions: Re-write the equation of
the line in slope-intercept form.
Slope must be in simplest/fraction
form. Do NOT put any spaces in
your answer.
Standard Form
2x+y=8
3x-y=7
5x+2y=8
4x+3y=9
X-y=0
x+4y=16
4x-3y=6
x+2y=-10
x+y=3
3x-2y=9
Answers
(Please vote me Brainliest if this helps!):
y = -2x + 8y = 3x - 7y = [tex]-\frac{5}{2}x[/tex] + 4y = [tex]-\frac{4}{3}x[/tex] + 3y = x + 0y = [tex]-\frac{1}{4}x[/tex] + 4y = [tex]\frac{4}{3}x[/tex] - 2y = [tex]-\frac{1}{2}x[/tex] - 5y = -x + 3y = [tex]\frac{3}{2}x - \frac{9}{2}[/tex]
Find the equivalent fraction of 2/3 having denominator 6 and 6 and numerator 10.
Answers
Step-by-step explanation:
having denominator 6,
X/6= 2/3
X= 4
so, 4/6= 2/3
having numerator 10,
10/X= 2/3
X= 15
so, 10/15= 2/3
write -9.38 as a mixed number
Answers
Answer:
-9/38/100=-9/19/50
hope it helps!!!!!
9. A line contains L(-4,-4) and M(2, 3). Line q is in
the same coordinate plane but does not intersect IM.
Line q contains point N.
Answers
don't know hahahahahahahau
Find the measure of one interior angle of a regular 16-gon.
Answers
Answer:
157.5
Step-by-step explanation:
14×180/16
= 315/2
= 157.5
Answer:
Step-by-step explanation:
16 sides
n-2=14
14x 180 = 2250
2250/16 =157.5 = 158
HELP PLEASSSEEE!!!???
(1 point) An advertiser goes to a printer and is charged $40 for 60 copies of one flyer and $50 for 170 copies of another
flyer. The printer charges a fixed setup cost plus a charge for every copy of single-page flyers. Find a function that
describes the cost of a printing job, if x is the number of copies made. C(2) =
HELPPPPPP PLEASSSEEE!!!
Answers
9514 1404 393
Answer:
C(x) = x/11 + 34 6/11
Step-by-step explanation:
The two given points are (x, C) = (60, 40) and (170, 50). These can be used in the 2-point form of the equation for a line to find the cost function.
y = (y2 -y1)/(x2 -x1)(x -x1) +y1
y = (50 -40)/(170 -60)(x -60) +40
y = (1/11)(x -60) +40
C(x) = 1/11x +34 6/11 . . . . . . cost as a function of copies made
Find the total amount paid on a loan when the monthly payment is $189 and the loan is paid off in 48 months. Use the formula A = MN, where A is the total amount paid, M is the monthly payment, and N is the number of payments.
Answers
Answer:
Step-by-step explanation:
M=189
N = 48
A = M*N
A = 189 * 48
A = 9072
Find the missing angle.
Answers
Answer:
Your answer is K= 60
I hope it's helps you
need help please answer
Answers
2
Stuart uses 3 of a cup of sugar for a recipe. How many cups of sugar are needed for 7 recipes?
Express the amount as a mixed number.
cups
Blank 1:
Blank 2:
Answers
Answer:
Blank1=4 blank2=2\3
Step-by-step explanation:
2\3 for 1 recipe
For 7 recipe = 7 x 2\3
= 14\2 x 2\3 -> take out the 2
14\3 = 14 divided by 3 is 4 with remainder of 2
So, 4 2\3
What is 2 2/15 - 1 2/3=
I ended up with 2 3/21
Really need help with this!
Answers
Answer:
[tex]2 \frac{2}{15} - 1 \frac{2}{3}[/tex]
[tex] = \frac{32}{15} - \frac{5}{3} [/tex]
[tex] = \frac{32 \times 1 - 5 \times 5}{15} [/tex]
[tex] = \frac{32 - 25}{15} [/tex]
[tex] = \frac{7}{15} [/tex]
4. Write the equation of the line that
passes through (9,4) (0-5)
Answers
Answer:
Equation of line is y = x - 5
Step-by-step explanation:
general equation of a line:
[tex]y = mx + c[/tex]
m » slope
c » y-intercept
for the slope, m :
[tex]slope = \frac{y_{2} - y_{1} }{x _{2} - x_{1} } [/tex]
substitute in accordance of the points:
[tex]slope = \frac{ - 5 - 4}{0 - 9} = \frac{ - 9}{ - 9} \\ \\ m= 1[/tex]
for y-intercept :
[ consider point (0, -5) ]
[tex]y = mx + c \\ - 5 = (0 \times 1) + c \\ c = - 5[/tex]
Therefore, equation of line is
[tex]{ \boxed{y = x - 5}}[/tex]
Draw a line segment of length 6.4 cm and divide it in to 4 equal parts. Write the length of each part. (Draw clearly)
Answers
Answer:
The line drawn has a length of 6.4 cm.
Also it is divided into 4 equal parts.
Hence let the measure of the each equal part be ' x '
So x * 4 parts = 6.4 cm
=> x = 6.4 cm / 4
=> x = 1.6 cm
Hence the measure of each equal part is 1.6 cm.
Hence solved !
Step-by-step explanation:
Ronnie sales wooden toothpicks it cost him $.10 to make a box of toothpicks he charges $1.20 per box toothpicks how many toothpicks does he need to sell to profit $1000
Answers
Answer:
910 boxes of toothpicks
Step-by-step explanation:
Create an equation to represent the situation, and let x represent the number of boxes of toothpicks.
1.20x - 0.1x = 1000
1.1x = 1000
x = 909.09
Round up to the nearest whole number:
= 910
So, Ronnie would have to sell 910 boxes of toothpicks to profit $1000.
I have picture of it
Answers
Answer:
basically multiply the prices by how much they filled up. That should give the answer.
Step-by-step explanation:
Wastewater is filling barrels at the rate of 11 quarts per hour. The recycling facility picks up 120 full barrels on each trip, and each barrel holds 12 quarts of wastewater. How many days are between each pickup?
Answers
Answer:
There are 6 days between each pick up days.
Step-by-step explanation:
the line y=mx+7 is perpendicular to the line y=3/4x-9. what is the value of m?
Answers
Answer:
-4/3
Step-by-step explanation:
In y=mx+b, m represents the slope. In the second line, we are multiplying x by (3/4). This represents the slope of the second line.
To find the slope of a perpendicular line, we take the negative reciprocal of the current slope. In the first line, m represents the slope. The reciprocal of 3/4 is 4/3, and the negative reciprocal of 3/4 is -4/3. This is the value of m
A falcon flies 300 kilometers per hour. A goose flies 140 kilometers per hour. If a falcon and a goose each fly for 2 hours, how many more kilometers will a falcon fly?
Answers
Answer:
320
Step-by-step explanation:
Distance covered by falcon in 2 hours=300*2=600 km
Distance covered by goose in 2 hours=140*2=280 km.
Falcon flew 600-280=320 km more than goose
PLssssssss helppppppppppppppppppppppp
Answers
Answer:
6 ft
Step-by-step explanation:
If other factors are held constant, how does sample size influence the likelihood of rejecting the null hypothesis and measures of effect size such as r^2 and Cohen's d?A) A larger sample increases both the likelihood and measures of effect size.B) A larger sample increases the likelihood but has little influence on measures of effect size.C) A larger sample decreases the likelihood but has little influence on measures of effect size.D) A larger sample decreases both the likelihood and measures of effect size.
Answers
Answer:
2000
Step-by-step explanation:
x/5=13
how do you solve for x
Answers
Answer:
X=65
Step-by-step explanation:
X/5=13
X=13×5
=65
hope it helps!!!
«
Use complete sentences to describe how simplifying expressions with multiple logarithms makes evaluating
expressions less complicated.
Answers
Step-by-step explanation:
bracket first in simplify
Saji swims 1/3 mi every day. One day he had already swum 1/5 mi. How much further should he swim?
Answers
9514 1404 393
Answer:
2/15 mi
Step-by-step explanation:
The difference of two fractions can always be computed using the pattern ...
[tex]\dfrac{a}{b}-\dfrac{c}{d}=\dfrac{ad-bd}{bd}[/tex]
Using this pattern to find the difference of 1/3 and 1/5, we see it is ...
1/3 -1/5 = (5 -3)/(3·5) = 2/15
Saji must swim 2/15 miles further to complete his 1/3 mile swim.
What is the slope of a line with the equation y-2=-3/4(x+5)?
Answers
Answer:
Gradient: m = -3/4
Step-by-step explanation:
Please view the PDF attached for full step by step explanation