# I Divergence/Convergence for Telescoping series

1. Mar 25, 2017

### 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.

2. Mar 25, 2017

### zwierz

that is right i guess

3. Mar 25, 2017

### zwierz

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

4. Mar 25, 2017

### Neon32

what's $a_n$?

5. Mar 25, 2017

### zwierz

a term of the series

6. Mar 25, 2017

### Neon32

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

7. Mar 25, 2017

### zwierz

8. Mar 25, 2017

### Neon32

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.

9. Mar 25, 2017

### zwierz

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

10. Mar 25, 2017

### Neon32

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

11. Mar 25, 2017

### zwierz

He restored the proof of my proposition for this concrete example