# I was wondering whether a sequence like$$x_n=n\sin • dalcde In summary, the conversation discussed the convergence or divergence of the sequence x_n=n+\sin n in the extended real number system. It was concluded that the sequence diverges, or converges to infinity. This is because the sequence becomes arbitrarily large as n increases, regardless of the coefficient in front of n. dalcde I was wondering whether a sequence like [tex]x_n=n\sin n$$
converges* to infinity or diverges.

I'm pretty sure it goes to infinity but it still oscillates.

*Let's say we are in the extended real number system where we can converge to infinity

EDIT: I mean
$$x_n=n+\sin n$$

Last edited:

first of all there is no such thing as convergence to infinity
and infinity is not really a number

converges is it goes towards a certain value like a limit.

Yea i agree with daclde.

The sin n part will just oscillate from -1 to 0 to 1

So yea it diverges . No clue about extended real system but

Divergence doesn't really have points where it must diverge towards. Rather anything that doesn't converge to a specific real point is defined as divergent. Divergence towards infinity is just a popular saying for when something grows without bound. Just like oscillations, divergence towards infinity is not heading towards any definite point.

In the extended real system, this sequence is still divergent as it oscillates between positive and negative.

Sorry, I wanted to say
$$x_n=n+\sin n.$$
Does it converge or diverge in this case?

dalcde said:
Sorry, I wanted to say
$$x_n=n+\sin n.$$
Does it converge or diverge in this case?

This will diverge (or converge to infinity in the extended reals). the reason is that n+sin(n) becomes arbitrarily large. That is, we have

$$n-1\leq n+sin(n)$$

and the sequence n-1 goes to infinity.

It converges to infinity in this case since n-1 is a lower bound.

edit: micromass explained it better

$$0.5n + \sin n$$
since the previous sequence is actually always increasing?

it still diverges

You can put any (positive) number you like in front of the "n". Eventually, n will be so large that even the product is far larger than sin(n).

## 1. What is the purpose of the "n" in the sequence?

The "n" in the sequence represents the position of the term in the sequence. It is a variable that starts at 1 and increases by 1 for each subsequent term.

## 2. How does the "sin" function affect the sequence?

The "sin" function, also known as the sine function, takes the value of "n" and outputs a corresponding value between -1 and 1. This value is then multiplied by "n", resulting in a sequence where the terms oscillate between positive and negative values.

## 3. Can the sequence be graphed?

Yes, the sequence can be graphed by plotting the value of "n" on the x-axis and the output of the "sin" function on the y-axis. The resulting graph will show a series of peaks and valleys, with the amplitude increasing as "n" increases.

## 4. Is there a pattern to the sequence?

Yes, there is a pattern to the sequence. As "n" increases, the terms in the sequence will oscillate between positive and negative values with a gradually increasing amplitude. This pattern continues indefinitely.

## 5. How is this sequence used in science?

This sequence is often used in physics and engineering to model oscillatory phenomena, such as the movement of a pendulum or a vibrating spring. It can also be used in mathematics to demonstrate the concept of a convergent series, where the terms of the sequence approach a fixed value as "n" increases.

Replies
12
Views
2K
Replies
13
Views
7K
Replies
6
Views
770
Replies
13
Views
1K
Replies
4
Views
2K
Replies
125
Views
17K
Replies
2
Views
1K
Replies
1
Views
1K
Replies
7
Views
1K
Replies
6
Views
2K