Convergence or Divergence of Series with Non-Convergent Sinusoidal Term

[Jbreezy]

Homework Statement

Test the series for convergence or divergence

Homework Equations

A_n = Ʃ 1/(2+sin(k)) from k = 1 to ∞

The Attempt at a Solution

I looked at this and I thought that sin(k) does not have a limit as k goes to infinity. So I was thinking that Lim k--> ∞ A_n = Does not exist. So, the series is divergent. I origonally thought to use the alternating series test since A_n is alternating but I didn't really get anywhere. How is my reasoning with this problem? Right track or no? Thanks,
J

 

[LCKurtz]

Homework Statement

Test the series for convergence or divergence

Homework Equations

A_n = Ʃ 1/(2+sin(k)) from k = 1 to ∞

The Attempt at a Solution

I looked at this and I thought that sin(k) does not have a limit as k goes to infinity. So I was thinking that Lim k--> ∞ A_n = Does not exist. So, the series is divergent. I origonally thought to use the alternating series test since A_n is alternating but I didn't really get anywhere. How is my reasoning with this problem? Right track or no? Thanks,
J

That is not an alternating series. Every term is positive. In the denominator you have $1\le 2+\sin(k)\le 3$. Although your limit argument is OK, you can also use this inequality to understimate your series with an obviously divergent one.

 

[Jbreezy]

Why isn't it alternating? I was thinking it was since sin(k) is cyclic. You know between -1 and 1.

$1\le 2+\sin(k)\le 3$ You mean I want something smaller that diverges?

 

[Office_Shredder]

Jbreezy, what is the definition of an alternating series?

 

[Jbreezy]

Alternates. A_n+1 < A_n and Limit n--> infinity is 0.

 

[LCKurtz]

Why isn't it alternating? I was thinking it was

Every term in the series is positive.

since sin(k) is cyclic. You know between -1 and 1.

It is true that $\sin k$ is between $-1$ and $1$. But there is nothing "cyclic" about it, whatever you mean by cyclic.

$1\le 2+\sin(k)\le 3$ You mean I want something smaller that diverges?

Yes, a smaller series which is divergent.

 

[LCKurtz]

JBreezy, if, after as long as you have been posting, you can't bring yourself to quote the message to which you are replying, our conversation is going to be very short.

 

[Office_Shredder]

Alternates. A_n+1 < A_n and Limit n--> infinity is 0.

No, an alternating series is one in which the sign of the terms changes every time n increases by 1. What you stated are the conditions required for the alternating series convergence test to be used, the conditions that are needed on top of the series being alternating.

So we go back to the question of: is this series alternating? Do the terms change sign every time? The answer to that is no: 1/(2+sin(k)) does not change signs every time k increases by 1.

 

[Jbreezy]

Yeah. Followed.