Divergence/Convergence for Telescoping series

[Neon32]

Can I use the divergence test on the partial sum of the telescoping series?
Lim n>infinity an if not equal zero then it diverges

The example below shows a telescoping series then I found the partial sum and took the limit of it. My question is shouldn't the solution be divergent? Since the result -1+cos 1 is not equal to 0? I'm confused.

 

[zwierz]

that is right i guess

 

[zwierz]

Lim n>infinity an if not equal zero then it diverges

it is $a_n$ must tend to zero, not the sum :)

 

[Neon32]

it is $a_n$ must tend to zero, not the sum :)

what's $a_n$?

 

[zwierz]

a term of the series

 

[Neon32]

a term of the series

How can I determine convergence/divergence for telescopinc series then?

 

[zwierz]

you have already done this

 

[Neon32]

you have already done this

I haven't? the solution is above, however I don't quite understand. I want a general rule to detect convergence/divergence fore telescoping series.

 

[zwierz]

The series $\sum (b_{n+1}-b_n)$ is convergent iff the sequence $b_n$ is convergent

 

[Neon32]

The series $\sum (b_{n+1}-b_n)$ is convergent iff the sequence $b_n$ is convergent

but he didn't find the convergence of bn sepeartly in the solution above?

 

[zwierz]

He restored the proof of my proposition for this concrete example