Does the series Ʃ[k=1,inf] tan(k)/(k^2+1) converge or diverge?

In summary, the conversation is about determining whether the series Ʃ[k=1,inf] tan(k)/(k^2+1) converges or diverges. The person asking the question is struggling to find a solution and is also curious about the possibility of having two intervals of convergence for a series. Another person suggests using the integral test and provides an argument for why the series may diverge.
  • #1
GreenPrint
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Homework Statement



Determine if the following series converges or diverges.

Ʃ[k=1,inf] tan(k)/(k^2+1)

Homework Equations





The Attempt at a Solution



I have no idea how to solve this problem. Now that I think of it, I have never solved a single question about series were I'm asked about convergence or divergence of a series with tangent or cotangent as part of the series. Tangent and cotangent are not defined at multiples of pi/2 excluding multiples of pi, but the series is from k to infinity were k is the set of integers and so the numerator all by itself would never go to positive or negative infinity at any k. Yet I can't seem to come up with a solution to this problem.

Also just a quick question. If I'm given a particular series in which I know the function which it represents, if the function is undefined at some given points, like for example 1/(2-x) or something of the sort, could I automatically conclude that the series doesn't converge at positive 2 sense the function doesn't?

Also I have never seen a problem were the interval of convergence included two intervals like [-10,5)(5,22] or something of the sort just [-10,5) if that makes any sense at all. Is it possible to have series were there are two intervals of convergence instead of just one?

Thank you for any help
 
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  • #2
=( ah man
 
  • #3
I posted a message about this series in the Homework Helpers' section. Maybe somebody there will have a good idea. The integral test seems like a way to go, but I haven't carried it all the way through.
 
  • #4
I believe this series diverges.

A condition for a series to converge is that it needs to be possible for any epsilon > 0, that there is an N'th term such that all subsequent terms have an absolute value less than epsilon.

tan(k) can get arbitrarily large as k comes arbitrarily close to pi/2 + m pi for some m.
However large you want it to be, you can get it for some k.
So you can make always find a term larger than epsilon.

However, I'm afraid my argument isn't completely rigorous yet.
 

FAQ: Does the series Ʃ[k=1,inf] tan(k)/(k^2+1) converge or diverge?

1. What is an infinite series in Calculus 2?

An infinite series in Calculus 2 is a sequence of terms that continues to add up infinitely. It is represented by the summation notation Σ and can have a finite or infinite number of terms.

2. How do you determine if an infinite series is convergent or divergent?

To determine if an infinite series is convergent or divergent, you can use various tests such as the comparison test, ratio test, or the integral test. These tests involve comparing the series to known convergent or divergent series, or integrating the series to analyze its behavior.

3. What is the difference between a convergent and divergent infinite series?

A convergent infinite series is one that has a finite sum, meaning that the sum of all the terms in the series approaches a specific value as the number of terms increases. A divergent infinite series, on the other hand, does not have a finite sum and the terms either approach infinity or oscillate between positive and negative values.

4. What is the significance of infinite series in real-world applications?

Infinite series have various applications in fields such as physics, engineering, and economics. They are used to model continuous phenomena, such as the motion of a pendulum or the growth of a population. They also play a crucial role in the development of mathematical concepts and theories.

5. How can I use infinite series to approximate a value?

Infinite series can be used to approximate a value by taking a finite number of terms and adding them up. The more terms that are included in the approximation, the closer the value will be to the actual value. This is known as the partial sum of an infinite series.

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