Studying the convergence of a series with an arctangent of a partial sum

In summary, the conversation discusses the convergence of a series and the reasoning behind it. One person believes the series is divergent while the other argues that it is absolutely convergent. The main difference lies in the estimation of a lower bound for the numerator. The conversation also touches on the difference between using a true statement and applying a false statement in a similar situation.
  • #1
Amaelle
310
54
Homework Statement
look at the image below
Relevant Equations
asymptotic behaviour, limit of partial sum
Greeting
I'm trying to study the convergence of this serie

1612345260662.png


I started studying the absolute convergence
because an≈n^(2/3) we know that Sn will be divergente S=∝ so arcatn (Sn)≤π/2 and the denominator would be a positive number less than π/2, and because an≈n^(2/3) and we know 1/n^(2/3) > 1/n so we conclude that this serie divergent (not absolutely convergent till this point)

but here is the solution of the book
1612345750480.png

so according to them the serie is absolutely convergent

I would be grateful if someone could explain me where I get wrong in my reasoning

Many thanks in advance!

Best
 
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  • #2
You don’t seem to investigate smallness of numerator as the answer does.
 
  • #3
thanks a lot for your prompt reply
I consider it as a postive number let's call it A 0<A<pi(2) A/n^2/3 > A/n and we know A/n is divergent so should be A/n^2/3 ?
where is the flaw in my reasoning?
 
  • #4
But what is A is really, really, really small?

If A=1/n for example, then A/n does converge.

You assumed there was some constant ##A>0## that was a lower bound for the numerator, but no such constant exists.
 
  • Informative
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  • #5
The solution estimates A as 1/Sn < 1/an
 
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  • #6
Office_Shredder said:
But what is A is really, really, really small?

If A=1/n for example, then A/n does converge.

You assumed there was some constant ##A>0## that was a lower bound for the numerator, but no such constant exists.
thanks alot
I want to ask about the difference between this case and between when we have for a example a dominator with sin(n) and we just say that |sin(n)|<1
what is the difference between this case and the exercice below?
many thanks in advance!
 
  • #7
Sure. ##|\sin(n)| < 1## is a true statement.

You want to do something like

##|\pi/2 - \arctan(n) | > 0.01##, where 0.01 is a number I just came up with (any number would work). But that's not a true statement, so you can't use it.
 
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  • #8
Office_Shredder said:
Sure. ##|\sin(n)| < 1## is a true statement.

You want to do something like

##|\pi/2 - \arctan(n) | > 0.01##, where 0.01 is a number I just came up with (any number would work). But that's not a true statement, so you can't use it.
thanks a million!
 

Related to Studying the convergence of a series with an arctangent of a partial sum

1. What is a series with an arctangent of a partial sum?

A series with an arctangent of a partial sum is a mathematical expression that involves the sum of a sequence of terms, where each term is the arctangent of a partial sum of the sequence. This type of series is commonly used in calculus and has applications in various fields of science and engineering.

2. How is the convergence of a series with an arctangent of a partial sum determined?

The convergence of a series with an arctangent of a partial sum can be determined using various convergence tests, such as the ratio test, the root test, and the comparison test. These tests help determine whether the series converges or diverges, and in some cases, can also provide information about the rate of convergence.

3. What are some real-world applications of studying the convergence of a series with an arctangent of a partial sum?

The study of the convergence of a series with an arctangent of a partial sum has various real-world applications, such as in the analysis of electrical circuits, the calculation of infinite sums in physics, and the approximation of functions in computer science. It is also used in the study of oscillatory systems and in the development of numerical methods for solving differential equations.

4. What are some common challenges when studying the convergence of a series with an arctangent of a partial sum?

One of the main challenges when studying the convergence of a series with an arctangent of a partial sum is finding a suitable convergence test to determine the convergence or divergence of the series. Another challenge is dealing with series that have alternating signs, as they require different convergence tests and may have more complex convergence behavior.

5. How does the convergence of a series with an arctangent of a partial sum relate to the overall convergence of a sequence?

The convergence of a series with an arctangent of a partial sum is closely related to the overall convergence of a sequence. In fact, the convergence of the series is determined by the convergence of the sequence of partial sums. If the sequence of partial sums converges, then the series also converges, and vice versa. This relationship is important in understanding the behavior of series and sequences in mathematical analysis.

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