Question about the convergence of a series

In summary: Statement A does not only fail where terms approaching ## n \to \infty ## may be negative: an example with strictly positive terms for both series was given in #3.
  • #1
Amaelle
310
54
Homework Statement
look at the image
Relevant Equations
Assymptotic approximation
Greetings!

I have a question about one assumption regarding this question even though I agree with the answer but I have a doubt about A, because
when we study the convergence of a serie we use the assymptotic approximation, so why A is not correct?
thank you!
when we
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  • #2
Exactly how is ##a_n## ~ ##b_n## defined? Is ##1/n## ~ ##1/n^2##?
 
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  • #3
FactChecker said:
Is ##1/n## ~ ##1/n^2##?

My Italian isn't great but I think in the context of the question this works better as: Does ## \frac 1 {n ^ 2} \sim \frac 1 n ## as ## n \to +\infty ##?
 
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  • #4
pbuk said:
My Italian isn't great but I think in the context of the question this works better as: Does ## \frac 1 {n ^ 2} \sim \frac 1 n ## as ## n \to +\infty ##?
I think they mean something like log (1+1/n) ≈ 1/n , this why I´m asking
 
  • #5
FactChecker said:
Exactly how is ##a_n## ~ ##b_n## defined? Is ##1/n## ~ ##1/n^2##?

This is the issue.

If it means [tex]
\lim_{n \to \infty} |a_n - b_n| \to 0[/tex] then indeed [itex]1/n \sim 1/n^2[/itex] and convergence of [itex]\sum a_n[/itex] tells us nothing about the convergence of [itex]\sum b_n[/itex].

However, to my mind [itex]a_n \sim b_n[/itex] as [itex]n \to \infty[/itex] means [tex]
\lim_{n \to \infty} \frac{|a_n - b_n|}{|a_n|} \to 0.[/tex] One can show that if [itex]a_n \sim b_n[/itex] in this sense then there exists [itex]N \in \mathbb{N}[/itex] such that [tex]
\left| \sum_{n=N}^M b_n - \sum_{n=N}^M a_n \right|
< \frac12 \sum_{n=N}^M |a_n|
[/tex] for all [itex]M \geq N[/itex], and it follows that if [itex]\sum a_n[/itex] converges absolutely then [itex]\sum b_n[/itex] converges absolutely. However I have not been able to show that if [itex]\sum a_n[/itex] converges but not absolutely then [itex]\sum b_n[/itex] converges, but nor have I found a counterexample.
 
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  • #6
Amaelle said:
I think they mean something like log (1+1/n) ≈ 1/n , this why I´m asking
Yes I can see that. The point is that ## a_n = \frac 1 {n ^ 2}, b_n = \frac 1 n ## is a counter-example that makes statement A incorrect because we have ## a_n \sim b_n ## as ## n \to +\infty ## and we have ## \Sigma a_n ## converging to ## 2 ## but ## \Sigma b_n = \Sigma \frac 1 n ## diverges.

This all assumes an interpretation of ## a_n \sim b_n ## similar to @pasmith's first one. I read it like this:
$$ a_n \sim b_n \text { as } n \to +\infty \iff \forall (\varepsilon > 0) \ \exists N : \forall (n > N) \ |a_n - b_n| < \varepsilon $$
 
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  • #7
pbuk said:
Yes I can see that. The point is that ## a_n = \frac 1 {n ^ 2}, b_n = \frac 1 n ## is a counter-example that makes statement A incorrect because we have ## a_n \sim b_n ## as ## n \to +\infty ## and we have ## \Sigma a_n ## converging to ## 2 ## but ## \Sigma b_n = \Sigma \frac 1 n ## diverges.

This all assumes an interpretation of ## a_n \sim b_n ## similar to @pasmith's first one. I read it like this:
$$ a_n \sim b_n \text { as } n \to +\infty \iff \forall (\varepsilon > 0) \ \exists N : \forall (n > N) \ |a_n - b_n| < \varepsilon $$
amazing explanation thanks a million!
 
  • #8
Amaelle said:
I think they mean something like log (1+1/n) ≈ 1/n , this why I´m asking
You need to determine exactly what the book with the problem means by '~'. If the book leaves it vague, then A is false. And if the book says that it means ##lim_{n\to\infty} |a_n-b_n| = 0##, A is false.
 
  • #9
I think the problem here is that in order to use the criteria for convergence the terms of the series must be NON negative,which is not assumed in statement A. Otherwise you would have to check for the absolute convergence of the series.
 
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  • #10
Davide97 said:
I think the problem here is that in order to use the criteria for convergence the terms of the series must be NON negative,which is not assumed in statement A. Otherwise you would have to check for the absolute convergence of the series.
thanks a million!
 
  • #11
Davide97 said:
I think the problem here is that in order to use the criteria for convergence the terms of the series must be NON negative,which is not assumed in statement A. Otherwise you would have to check for the absolute convergence of the series.
Statement A does not only fail where terms approaching ## n \to \infty ## may be negative: an example with strictly positive terms for both series was given in #3.
 
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  • #12
I see now, thanks a million for your feedback!
 

FAQ: Question about the convergence of a series

1. What is the definition of convergence for a series?

The convergence of a series is a mathematical concept that refers to the behavior of the terms in a series as the number of terms approaches infinity. A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases.

2. How do you determine if a series converges or diverges?

There are several tests that can be used to determine the convergence or divergence of a series, such as the ratio test, the root test, and the integral test. These tests involve analyzing the behavior of the terms in the series and comparing them to known patterns of convergence or divergence.

3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series in which the sum of the absolute values of its terms is finite. Conditional convergence, on the other hand, refers to a series in which the sum of its terms is finite, but the sum of the absolute values of its terms is infinite.

4. Can a series converge to more than one value?

No, a series can only converge to a single value. If a series converges to more than one value, it is considered to be divergent.

5. How does the convergence of a series relate to its rate of growth?

The convergence of a series is closely related to its rate of growth. A series with a slower rate of growth is more likely to converge, while a series with a faster rate of growth is more likely to diverge. However, this is not always the case, and it is important to use convergence tests to determine the behavior of a series.

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