Converge absolutely or conditionally, or diverges?

  • Thread starter rcmango
  • Start date
In summary, the series in question can be determined to converge conditionally by using the Leibniz test for alternating series and verifying three conditions. The function tan(1/n) is decreasing and as n-> infinity, it goes to 0.
  • #1
rcmango
234
0

Homework Statement



Determine if the series converges absolutely, converges conditionally, or diverges.

equation is here: http://img409.imageshack.us/img409/7353/untitledly5.jpg

Homework Equations



maybe alternating series, or harmonic series?

The Attempt at a Solution



not real familiar with tan with series.
haven't tried much, need supporting work for the answer.
need help.
 
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  • #2
As n-> infinity, tan(1/n) -> tan(0) -> 0
Does this help?
 
  • #3
Actually, this says nothing at all about the series. The implication is one way only: "Sum a_n converges ==> a_n-->0" but "a_n-->0 ==> nothing".

Actually the series satisfies all the criteria corresponding to the convergence of an alternating series. Remains to see if it converges absolutely. I.e. does

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

??
 
  • #4
no it doesn't converge absolutely because it continues on to infinity.

however, i do ask, how do you know to test it to be less than infinity? in other words, the convergence for a alternating series passes. but what other series convergence did not pass?

so ultimately, this will converge conditionally.

for my work, i could prove this by showing the alternating series? and then showing that it also continues on to infinity?

thanks again for all the help so far.
 
  • #5
What do you mean by "continues on to infinity" ?
 
  • #6
rcmango, i think you mean using the Leibniz test (for alternating series)
there are three conditions, check all to prove.
 
  • #7
chanvincent said:
As n-> infinity, tan(1/n) -> tan(0) -> 0
Does this help?
It is more to the point that tan(1/n) is decreasing. Knowing that it goes to 0 is neither necessary nor helpful.
 

FAQ: Converge absolutely or conditionally, or diverges?

1. What is absolute convergence?

Absolute convergence is a type of convergence in which the sum of the absolute values of the terms in a series converges. This means that the series will converge regardless of the order in which the terms are added.

2. What is conditional convergence?

Conditional convergence is a type of convergence in which the sum of the terms in a series converges, but the sum of the absolute values of the terms diverges. This means that the series will only converge if the terms are added in a specific order.

3. How do you determine if a series converges absolutely or conditionally?

To determine if a series converges absolutely or conditionally, you can use the ratio or root test. If the limit of the absolute value of the terms is less than 1, the series converges absolutely. If the limit is equal to 1, the series may converge conditionally, and if the limit is greater than 1, the series diverges.

4. What is an alternating series?

An alternating series is a series in which the terms alternate in sign, meaning that every other term is positive or negative. These types of series often involve alternating between addition and subtraction, and can be tested for convergence using the alternating series test.

5. What does it mean if a series diverges?

If a series diverges, it means that the sum of its terms does not have a finite value. This can happen if the terms increase without bound or if they oscillate between positive and negative values without approaching a specific value. Divergence can also occur if the series fails the convergence tests, such as the ratio or root test.

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