Convergent Telescoping Series: \sum_{n=1}^{\infty} \arctan(n+1)-\arctan(n)

[courtrigrad]

\sum_{n=1}^{\infty} \arctan(n+1)-\arctan(n).

So we want to take \lim s_{n} = \lim_{n\rightarrow \infty} (\arctan 2-\arctan 1)+(\arctan 3-\arctan 2) + ... + (\arctan(n+1)-\arctan n). From this I can see all of the terms cancel except \arctan 1. But then how do we get: \lim_{n\rightarrow \infty} \arctan(n+1)-\arctan 1? Wouldn't the \arctan(n+1) cancel out?

Thanks

 

[courtrigrad]

I got it. I was assuming that it would keep on going. But it stops. I have to take the limit. so \arctan(n+1) and a -\arctan 1 are left.