Lim (n→∞)((-1)^n)/n: Does it Converge or Diverge?

  • Thread starter soul
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In summary, the question asks if the sequence and series lim(n->infinity)((-1)^n)/n converge, and the answer is yes. This can be shown using various tests for convergence such as the alternating series test and Dirichlet's test.
  • #1
soul
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I know it is easy but I need help. lim(n->infinity)((-1)^n)/n diverge or converge?and why?

p.s: question may not be understandable. Sorry for that.
 
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  • #2
what are your thoughts?? Show what you did so far!
 
  • #3
there's a test for this type of series
 
  • #4
Well, in fact I did not do so much. I could not decide whether it converges 1 or -1 or since we do not know where it converges if we can simply say that it is diverge.
 
  • #5
why in the world would you think it converges to 1 or -1
 
  • #6
I was talking about (-1)^n, when I said it converges 1 or -1.
 
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  • #7
lim(n->infinity)((-1)^n)
----------------------
lim(n->infinity)n

Couldn't you just evaluate the series this way?
 
  • #8
Feldoh said:
lim(n->infinity)((-1)^n)
----------------------
lim(n->infinity)n

Couldn't you just evaluate the series this way?

And get what?
 
  • #9
soul said:
I know it is easy but I need help. lim(n->infinity)((-1)^n)/n diverge or converge?and why?

p.s: question may not be understandable. Sorry for that.
Are you just looking whether that limit exists, or your original question requires sth else, like whether the series with the general term a_n=(-1)^n /n converges or not?

WHat is your original question.?
 
  • #10
This is the original question. If you want to check it, you can look at p.757 of Thomas's calculus (question 24). It says that find whether this sequence is convergent or not. If convergent find the limit of it?
 
  • #11
soul said:
This is the original question. If you want to check it, you can look at p.757 of Thomas's calculus (question 24). It says that find whether this sequence is convergent or not. If convergent find the limit of it?
Well, to show that a series is convergent you first need to determine whether it is monotonic or not, then show that it is bounded. Or if the limit of that sequence exists then from a theorem we conclude that it is convergent also.
To evaluate that limit you might want to breake the problem into two parts.
First let n=2k, where k is any integer, and see where does that sequence tend to as k-->infinity.
Second take n=2k+1, or n=2k-1, where again k is any integer, and see where does that sequence tend to, as k-->infinity. If they match then the lim exists, so the sequence converges to that value.
there are other ways to show this though. Others might present you to those ways!
 
  • #12
Or I assume you could use a test for convergence
 
  • #13
soul said:
I know it is easy but I need help. lim(n->infinity)((-1)^n)/n diverge or converge?and why?

p.s: question may not be understandable. Sorry for that.

Are you asking if this converges:

[tex]\lim_{n \to \infty} \frac{(-1)^n}{n}[/tex]

Look at the first few terms:

[tex]\frac{-1}{1}, \frac{1}{2}, \frac{-1}{3}, \frac{1}{4}, \ldots[/tex]

Do those numbers tend to get close to something?

Other people here seem to ask if you're wondering if this converges:

[tex]\sum_{n=1}^\infty \frac{(-1)^n}{n}[/tex]

Have you looked at the first few terms of that?
 
  • #14
An alternating series will converge if it deceases at least as quickly as 1 over n. (see alternating series test)
 
  • #15
1/n remains always positive but tends to 0 as n tends to infinity.
Using Leibnitz's test, convergence of the series you mentioned follows.

Or you could apply the Dirichlet's test.
The sequence of partial sums of (-1)^n is bounded, and 1/n remains always positive but tends to 0 as n tends to infinity.
So convergence follows naturally.
 
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Related to Lim (n→∞)((-1)^n)/n: Does it Converge or Diverge?

1. What is the formula for Lim (n→∞)((-1)^n)/n?

The formula is Lim (n→∞)((-1)^n)/n.

2. How is (-1)^n defined in this formula?

In this formula, (-1)^n is an alternating sequence that switches between positive and negative values as n increases.

3. What does it mean for a sequence to converge?

A sequence converges if its terms approach a single finite value as n increases without bound. It can also be thought of as the sequence "settling" towards a specific value.

4. Does Lim (n→∞)((-1)^n)/n converge or diverge?

The sequence Lim (n→∞)((-1)^n)/n does not converge because the terms do not approach a single finite value as n increases without bound. Instead, it oscillates between positive and negative values.

5. What is the significance of (-1)^n in this formula?

The (-1)^n in this formula represents the alternating nature of the sequence. It is responsible for the sequence not converging and instead oscillating between positive and negative values.

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