Convergence of the Sum of Tan(1/n)/(1+n) for n=1 to Infinity?

  • Thread starter kskiraly
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In summary, the sum of tan(1/n)/(1+n) for n=1 to infinity can be evaluated by using the ratio test and the comparison/limit comparison tests. It can be seen that the fraction will continue to get smaller and smaller, and will always be smaller than 1/n. By comparison, since 1/n converges, tan(1/n)/(1+n) must also converge. However, it is important to consider error estimates for sin(1/n) and avoid comparing to the diverging harmonic series. Additionally, it should be noted that the convergence of the series in question is dependent on the condition that r > 1 for n^r.
  • #1
kskiraly
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Homework Statement



sum of tan(1/n)/(1+n) for n=1 to infinity

Homework Equations





The Attempt at a Solution



I tried using the ratio test and the comparison/limit comparison tests but can't think on anything to compare it to.
 
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  • #2
I'm far from a math wiz, but here's how I see it:tan(x) will always be between -1 and 1,

but tan (1/n) will always be positive so it will be between 0 and 1

(well, actually for this problem, the max value of the numerator is tan(1) and it continues to decrease)

and the denominator will be at the very least 2 and it will continue to increase

so your fraction will continue to get smaller and smaller

in fact, it will always be smaller than 1/n

since the top will be at most 1 and the the denominator will be greater than n

so, tan(1/n)/(n+1) < 1/nand you know that 1/n converges...so by comparison, tan(1/n)/(1+n) must also converge
 
Last edited:
  • #3
mybsaccownt said:
I'm far from a math wiz, but here's how I see it:


tan(x) will always be between -1 and 1,

Are you sure about that? What does tan(x) approach as x approaches pi/2 from the left?
 
  • #4
mybsaccownt said:
and you know that 1/n converges...so by comparison, tan(1/n)/(1+n) must also converge

The SEQUENCE with general term 1/n converges, does the series?
 
  • #5
Well.. sin(1/n) is approximately 1/n for large n. I would do some sort of error estimate for tan(1/n) = 1/n + error to see if the the series CONVERGES absolutely. (Error estimates for sin(1/n) may suffice.)

Watch out for comparing to the famously DIVERGING harmonic series.
 
  • #6
d_leet, ok tan(x) does approach infinity when cos(x) approaches 0

and

"The harmonic series diverges, albeit slowly, to infinity"

http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)

well...ok, I forgot about the n^r r > 1 condition for convergence

but that's why I put the disclaimer about not being a math wiz :-pbut...BUT...the series in question does converge

<---has super maple skills
 
  • #7
Thanks so much but doesn't 1/n diverge by p-series because p=1. remember p-series diverges where 1/(n^p) p<=1
 
  • #8
mybsaccownt sorry about the last post i didn't fully read your last one.
 

1. What does it mean for a series to converge or diverge?

Convergence and divergence refer to the behavior of a series, which is a sum of an infinite sequence of numbers. A convergent series is one where the terms of the sequence approach a finite limit as the number of terms increases. Conversely, a divergent series is one where the terms of the sequence do not approach a finite limit, but instead grow larger and larger.

2. How can I tell if a series will converge or diverge?

To determine the convergence or divergence of a series, you can use various tests such as the geometric series test, the comparison test, or the ratio test. These tests evaluate the behavior of the terms in the series and can help determine if the series will converge or diverge.

3. What is the significance of a convergent or divergent series?

A convergent series has a finite sum, which means that the terms in the series add up to a specific value. This can have practical applications in fields such as finance and engineering. A divergent series, on the other hand, does not have a finite sum and can be used to represent quantities that grow without bound, such as population growth or radioactive decay.

4. Can a series converge and diverge at the same time?

No, a series can either converge or diverge, but not both. If a series converges, it cannot diverge, and vice versa. However, it is possible for a series to be neither convergent nor divergent, in which case it is said to be oscillatory.

5. How does the behavior of the terms in a series affect its convergence or divergence?

The behavior of the terms in a series is crucial in determining its convergence or divergence. For example, if the terms in a series decrease in size and approach zero, the series is more likely to converge. On the other hand, if the terms in a series grow without bound, the series is more likely to diverge. Additionally, the rate at which the terms grow or decrease can also impact the convergence or divergence of a series.

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