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
vekshin1
3 years ago
12

Answer the question with explanation;​

Mathematics
2 answers:
PSYCHO15rus [73]3 years ago
7 0

Answer:

The statement in the question is wrong. The series actually diverges.

Step-by-step explanation:

We compute

\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

Therefore, by the series divergence test, the series \sum_{n=1}^\infty\frac{n^2}{(n+1)^2} diverges.

EDIT: To VectorFundament120, if (x_n)_{n\in\mathbb N} is a sequence, both \lim x_n and \lim_{n\to\infty}x_n are common notation for its limit. The former is not wrong but I have switched to the latter if that helps.

elena-s [515]3 years ago
7 0

Answer:

\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} = \text{div}

General Formulas and Concepts:

<u>Calculus</u>

Limits

  • Special Limit Rule [Coefficient Power Method]:                                         \displaystyle \lim_{x \to \pm \infty} \frac{ax^n}{bx^n} = \frac{a}{b}

Series Convergence Tests

  • nth Term Test:                                                                                               \displaystyle \sum^{\infty}_{n = 1} a_n \rightarrow \lim_{n \to \infty} a_n
  • Integral Test:                                                                                                 \displaystyle \sum^{\infty}_{n = a} f(n) \rightarrow \int\limits^{\infty}_a {f(x)} \, dx
  • P-Series:                                                                                                         \displaystyle \sum^{\infty}_{n = 1} \frac{1}{n^p}
  • Direct Comparison Test (DCT)
  • Limit Comparison Test (LCT)
  • Alternating Series Test (AST)
  • Ratio Test:                                                                                                     \displaystyle \sum^{\infty}_{n = 0} a_n \rightarrow \lim_{n \to \infty} \bigg| \frac{a_{n + 1}}{a_n} \bigg|

Step-by-step explanation:

*Note:

Always apply the nth Term Test as the first test to use for convergence.

Rules:

  1. If  \displaystyle  \lim_{n \to \infty} S_n = 0, then the nth Term Test is inconclusive.
  2. If  \displaystyle  \lim_{n \to \infty} S_n = l  (some number <em>l</em>), then the series is divergent by the nth Term Test.

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2}

<u>Step 2: Find Convergence</u>

  1. Substitute in variables [nth Term Test]:                                                       \displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} \rightarrow \lim_{n \to \infty} \frac{n^2}{(n + 1)^2}
  2. Expand:                                                                                                         \displaystyle \lim_{n \to \infty} \frac{n^2}{(n + 1)^2}= \lim_{n \to \infty} \frac{n^2}{n^2 + 2n + 1}
  3. Evaluate limit [Special Limit Rule - Coefficient Power Method]:                 \displaystyle \lim_{n \to \infty} \frac{n^2}{(n + 1)^2} = 1
  4. Compute [nth Term Test]:                                                                             \displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} = \text{div}

∴ by the nth Term Test, the series diverges.

Topic: AP Calculus BC (Calculus I + II)

Unit: Convergence Tests

You might be interested in
Point C'(-4,-3)C
Arturiano [62]

Answer:

i very diffucult question

6 0
3 years ago
Solve the inequality 7(x + 4) &gt; 0.<br> a.x -4<br> c.x 4
sesenic [268]
Solve your inequality step-by-step.

7(x+4)>0

Simplify both sides of the inequality.

7x+28>0

Subtract 28 from both sides
.
7x+28−28>0−28

7x>−28

Divide both sides by 7.

7x/7 > −28/7

x > −4

5 0
3 years ago
Read 2 more answers
Find the correct plugged in formula for each figure.
r-ruslan [8.4K]

Answer:

3

Step-by-step explanation:

I'm just guessing because you don't give any numbers or anything at all

4 0
3 years ago
Read 2 more answers
What is the y coordinate of the solution 3x+2y=12
Tomtit [17]

Answer:

12

Step-by-step explanation:

3x+2y=12

2y=-3x+12

y=-3/2+12

Honestly you dont have to do any of this work but i did it anyways, The y coordinate is always at the end of the point intercept form (y=mx+b form)

5 0
3 years ago
What is a solution to the equation 5x + 2y = -1?
Hunter-Best [27]
The answer to the solution of the equation 5x+2y=-1 is 1,-3)
5 0
3 years ago
Other questions:
  • I need help with adding and subtracting fractions
    14·2 answers
  • (8z^8/4z^4)^-4 please help me to solve this
    8·1 answer
  • How many natural numbers less than 300 are neither multiples of 2 nor multiples of 3?
    7·1 answer
  • Would a teaspoon be a good way to measure the capacity of a milk carton? Explain.
    12·1 answer
  • Need help please help
    13·1 answer
  • Select the solutions to this quadratic equation:
    10·1 answer
  • Find the area of this semi-circle with diameter 18<br> Give your answer rounded to 2 DP.
    5·1 answer
  • *insert chuck and his preggo sister with the gooey stuff coming out of her while the ded baby slides out*
    9·2 answers
  • How do you multiply ​
    12·1 answer
  • Fill in the blank:
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!