Help understanding the conditions of the Alternating Series

In summary: I understand correctly, the two cases would be divergent. If the first condition were to hold, then the series would converge. However, since that condition is not always met, the series diverges.
  • #1
dan38
59
0

Homework Statement


Okay from what I have learned to prove that a series converges via the alternating test, you must prove the following conditions

Homework Equations


1) an > 0
2) lim an (n--> infinity) = 0
and
3) a(n+1) < an

The Attempt at a Solution


However recently I've been encountering questions in my textbook where the first condition isn't filled, but the others two are and the answer is that it is converging
Doing further research online, I've found that most people simply disregard the first condition as well, i.e.
ww.youtube.com/watch?v=8qhVGeCkgGg

So Would anyone be able to clarify why the first condition is seemingly irrelevant?
 
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  • #2
dan38 said:
However recently I've been encountering questions in my textbook where the first condition isn't filled, but the others two are and the answer is that it is converging
Post those questions and your attempts. :smile: It's always easier to explain and find your mistakes when facing the questions themselves.
 
  • #3
ttp://www.wolframalpha.com/input/?i=-%28%28-1%29^n%29%2F%28n%2B1%29+convergence

Wouldnt you expect that to diverge since
an = -1/(n+1)
which isn't greater than 0 for all values of n
 
  • #4
Don't you mean: ##a_n = \frac{(-1)^n}{(n+1)}##? That would be wrong.
Use LaTeX to write it or just take a screenshot or picture of the whole question and attach it to your post.
 
  • #5
no that would be series itself
an = -1/(n+1)
i.e. the co-efficient of -1^n
so how would that converge??

dont quite know how to use latex yet sorry
 
  • #6
dan38 said:

Homework Statement


Okay from what I have learned to prove that a series converges via the alternating test, you must prove the following conditions

Homework Equations


1) an > 0
2) lim an (n--> infinity) = 0
and
3) a(n+1) < an

The Attempt at a Solution


However recently I've been encountering questions in my textbook where the first condition isn't filled, but the others two are and the answer is that it is converging
Doing further research online, I've found that most people simply disregard the first condition as well, i.e.
ww.youtube.com/watch?v=8qhVGeCkgGg

So Would anyone be able to clarify why the first condition is seemingly irrelevant?

You need to present your notation more carefully (as written, your statements look all wrong), but if I understand your question correctly, you are worried about some series being convergent event though some of the convergence test conditions you were taught do not apply. That is simply because the conditions you were taught are SUFFICIENT, not necessary. In other words, if such-and-such conditions hold we have a convergent series. This does not say anything at all about what happens when the conditions do NOT hold.

RGV
 
  • #7
so if it doesn't hold, do I use another test?
 
  • #8
dan38 said:
so if it doesn't hold, do I use another test?

Give a specific example. To try to speak in generalities at this stage is dangerous: it might leave you "knowing" things that are wrong.

RGV
 
  • #9
where my series = the negative of this

(-1)^n
--------
(n+1)
 
  • #10
sorry just had another quick question, if i had the negative series =

(-1)^n
--------
n^0.5


Could I use the comparison test to the modulus of this and then use the p-series to determine if the 1/root(n) converges/diverges?
 
  • #11
dan38 said:
where my series = the negative of this

(-1)^n
--------
(n+1)

So (assuming we start at n = 0), one series is
[tex] 1 -\frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots,[/tex]
while the other is
[tex] -1 + \frac{1}{2}- \frac{1}{3} + \frac{1}{4} - \cdots.[/tex]
Do you honestly and truly not see that convergence/divergence answers are exactly the same in the two cases?

RGV
 

1. What is an alternating series?

An alternating series is a mathematical series in which the terms alternate in sign, such as -1, 2, -3, 4, -5... These types of series are commonly used in calculus and can help to approximate the value of a function.

2. What are the conditions for an alternating series to converge?

The conditions for an alternating series to converge are that the absolute value of the terms in the series must decrease as the series progresses and must approach zero, and that the limit of the terms must be equal to zero as well.

3. How do you determine the convergence or divergence of an alternating series?

To determine the convergence or divergence of an alternating series, you can use the alternating series test or the ratio test. The alternating series test states that if the conditions for convergence are met, then the series will converge. The ratio test compares the ratio of consecutive terms and if the limit of this ratio is less than 1, the series converges.

4. Can an alternating series diverge?

Yes, an alternating series can diverge if the conditions for convergence are not met. For example, if the terms do not decrease in absolute value as the series progresses, or if the limit of the terms is not equal to zero, the series will diverge.

5. How can understanding alternating series be useful in real-life applications?

Understanding alternating series can be useful in real-life applications such as approximating the value of a function or solving problems involving oscillating quantities. It can also be applied in various fields of science such as physics, engineering, and economics.

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