Convergence of the series: Show ∑tan(π/4n) does not converge.

[Morragan]

Hey!
I have to prove, that the series does not converge, using a comparative criterion, but I have no idea how:
∑tan(π/4n)
All I could come up with is removing n, since tangent's value decreases, as it approaches 0 from the right, but it leads me nowhere.:???:

 

[stapel]

I have to prove that the series does not converge, using a comparative criterion, but I have no idea how:
∑tan(π/4n)

What is the summation? From n = 1 to infinity? So the summation is as follows?

. . . . .\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\, \tan\left(\dfrac{\pi}{4n}\right)\)

All I could come up with is removing n, since tangent's value decreases, as it approaches 0 from the right, but it leads me nowhere.

What do you mean by "removing n"? Please reply showing your work and reasoning. Thank you!

 

[Morragan]

What is the summation? From n = 1 to infinity? So the summation is as follows?
. . . . .\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\, \tan\left(\dfrac{\pi}{4n}\right)\)

Yes, yes it is

What do you mean by "removing n"? Please reply showing your work and reasoning. Thank you!

Well, to tell, if the sum does or does not converge, I need to find different sum, which value is either smaller or greater and state, whether is does or does not converge.
As I tried to prove, that it does converge, I stated, that for every n ∈ N, the value of tan(pi/4n) is lesser or equal to tan(pi/4), but it lead me nowhere.
I've used WolframAlpha, but it stated, that the sum does not converge, but I have no idea how to prove it.

 

[Morragan]

Ok, A friend of mine told me, that the key to understanding it is the fact, that tan(x) >= x and I am able to solve the problem with it, but how do I tell if that's true for every x ∈ [0 ; pi/2)?

 

[Steven G]

Ok, A friend of mine told me, that the key to understanding it is the fact, that tan(x) >= x and I am able to solve the problem with it, but how do I tell if that's true for every x ∈ [0 ; pi/2)?

What is the taylor expansion for tan(x)????

 

[pka]

Ok, A friend of mine told me, that the key to understanding it is the fact, that tan(x) >= x and I am able to solve the problem with it, but how do I tell if that's true for every x ∈ [0 ; pi/2)?

Define \(\displaystyle g(x)=\tan(x)-x\) can you say anything about \(\displaystyle g'(x)\) on \(\displaystyle \left[0,\frac{\pi}{4}\right]~?\)

 

[Morragan]

Ok, with your help, I got it right.
Thanks a lot!