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4 years ago

Which statement describes the sequence defined by

Mathematics

2 answers:

Llana [10]4 years ago

7 0

Answer:

a) The given sequence is diverges

$\lim_{n \to \infty} a_n$ = ∞

Step-by-step explanation:

Explanation

Given that the $n^{th}$ term sequence

$a_{n} = \frac{n^{3} - n}{n^{2} + 5 n}$

we have to prove that a given sequence converges or diverges

$\lim_{n \to \infty}a_{n} = \lim_{n \to \infty} \frac{n^{3} - n}{n^{2} + 5 n}$

$= \lim_{n \to \infty} \frac{n^{3}(1- \frac{1}{n^{3} }) }{n^{2}(1+ \frac{5n}{n^{2} } )}$

$= \lim_{n \to \infty} \frac{n(1- \frac{1}{n^{3} }) }{(1+ \frac{5n}{n^{2} } )}$

= ∞

$\lim_{n \to \infty} a_n$ = ∞

The given sequence is diverges

Schach [20]4 years ago

6 0

Answer:

it is A

Step-by-step explanation:

got it right on edu 2021

Read more about domain at:

brainly.com/question/24302079

4 0

3 years ago

The diagram shows how cos θ, sin θ, and tan θ relate to the unit circle. Copy the diagram and show how sec θ, csc θ, and cot θ r

DIA [1.3K]

Copy the diagram and show how sec θ, csc θ, and cot θ relate to the unit circle.

The representation of the diagram is shown if Figure 1. There's a relationship between sec θ, csc θ, and cot θ related the unit circle. Lines green, blue and pink show the relationship.

a.1 First, find in the diagram a segment whose length is sec θ.

The segment whose length is sec θ is shown in Figure 2, this length is the segment $\overline{OF}$ , that is, the line in green.

a.2 Explain why its length is sec θ.

We know these relationships:

(1) $sin \theta=\frac{\overline{BD}}{\overline{OB}}=\frac{\overline{BD}}{r}=\frac{\overline{BD}}{1}=\overline{BD}$

(2) $cos \theta=\frac{\overline{OD}}{\overline{OB}}=\frac{\overline{OD}}{r}=\frac{\overline{OD}}{1}=\overline{OD}$

(3) $tan \theta=\frac{\overline{FD}}{\overline{OC}}=\frac{\overline{FC}}{r}=\frac{\overline{FC}}{1}=\overline{FC}$

Triangles ΔOFC and ΔOBD are similar, so it is true that:

$\frac{\overline{FC}}{\overline{OF}}= \frac{\overline{BD}}{\overline{OB}}$ 

∴ $\overline{OF}= \frac{\overline{FC}}{\overline{BD}}= \frac{tan \theta}{sin \theta}= \frac{1}{cos \theta} \rightarrow \boxed{sec \theta= \frac{1}{cos \theta}}$ 

b.1 Next, find cot θ

The segment whose length is cot θ is shown in Figure 3, this length is the segment $\overline{AR}$ , that is, the line in pink.

b.2 Use the representation of tangent as a clue for what to show for cotangent.

It's true that:

$\frac{\overline{OS}}{\overline{OC}}= \frac{\overline{SR}}{\overline{FC}}$

But:

$\overline{SR}=\overline{OA}$
$\overline{OS}=\overline{AR}$

Then:

$\overline{AR}= \frac{1}{\overline{FC}}= \frac{1}{tan\theta} \rightarrow \boxed{cot \theta= \frac{1}{tan \theta}}$

b.3 Justify your claim for cot θ.

As shown in Figure 3, θ is an internal angle and ∠A = 90°, therefore ΔOAR is a right angle, so it is true that:

$cot \theta= \frac{\overline{AR}}{\overline{OA}}=\frac{\overline{AR}}{r}=\frac{\overline{AR}}{1} \rightarrow \boxed{cot \theta=\overline{AR}}$

c. find csc θ in your diagram.

The segment whose length is csc θ is shown in Figure 4, this length is the segment $\overline{OR}$ , that is, the line in green.