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AleksAgata [21]
3 years ago
14

The difference between a number and 12 is 20 Write an equation

Mathematics
1 answer:
Scorpion4ik [409]3 years ago
7 0
Assume that this number is n.
Difference between two number means the subtraction between them.
Therefore, "<span>The difference between a number and 12 is 20" means that when subtracting these two numbers, the result is 20.

Since the question did not mention which number is greater, therefore, we have two ways to write the equation. 
These two ways are as follows:
x - 12 = 20, and
12 - x = 20

</span>
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What is the limit of the infinite series?<br> ∞∑n=1 (3n^5 / 6n^6 + 1)
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Answer:

The limit of the infinite series is equal to zero.

The nth term test is inconclusive ∵ the limit is equal to 0.

By the Comparison Test, this sum diverges.

General Formulas and Concepts:
<u>Calculus</u>

Limits

  • Limit Rule [Variable Direct Substitution]:                                                 \displaystyle \lim_{x \to c} x = c

Series Comparison Tests

  • nth Term Test
  • Direct Comparison Test (DCT)

Step-by-step explanation:

<u>Step 1: Define</u>

\displaystyle \sum^{\infty}_{n = 1} \frac{3n^5}{6n^6 + 1}

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

  1. [Series] Define:                                                                                            \displaystyle a_n = \frac{3n^5}{6n^6 + 1}
  2. [Series] Set up [nth Term Test]:                                                                   \displaystyle \sum^{\infty}_{n = 1} \frac{3n^5}{6n^6 + 1} \rightarrow \lim_{n \to \infty} \frac{3n^5}{6n^6 + 1}
  3. [nth Term Test] Evaluate limit [Limit Rule - VDS]:                                       \displaystyle \lim_{n \to \infty} \frac{3n^5}{6n^6 + 1} = 0
  4. [nth Term Test] Determine Conclusiveness:                                              \displaystyle 0 ,\ \sum^{\infty}_{n = 1} \frac{3n^5}{6n^6 + 1} \ \text{can't be concluded that it converges}

Therefore, the nth term test is inconclusive and another test must be done.

<u>Step 3: Find Convergence Pt. 2</u>

  1. [DCT] Condition 1 [Define comparing series]:                                            \displaystyle \frac{1}{2} \sum^{\infty}_{n = 1} \frac{1}{n}
  2. [DCT] Condition 1 [Test convergence of comparing series]:                   \displaystyle p = 1 ,\ \frac{1}{2} \sum^{\infty}_{n = 1} \frac{1}{n} \ \text{divergent by p-series (harmonic series)}

∴ since the comparison series is divergent, then our original series is also divergent according to the Direct Comparison Test.

---

Learn more about limits: brainly.com/question/26091024

Learn more about Taylor Series: brainly.com/question/23558817

Topic: AP Calculus BC (Calculus I + II)

Unit: Taylor Series

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An expression is defined as a set of numbers, variables, and mathematical operations. The factored form of the expression 16x²+8x+32 is 8(2x²+x+4).

<h3>What is an Expression?</h3>

In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.

The completely factored form of the expression can be written as,

16x^2+8x+32\\\\=8(2x^2+x+4)

Thus, the factored form of the expression 16x²+8x+32 is 8(2x²+x+4).

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