Expert: Steve Holleran - 5/2/2007

Question
Hi, I wanted to know how to go about solving this infinite series problem by hand.  I've never encountered this type of problem before so I am not sure what to do to solve it.  The question asks to find the sum of this infinite series:

[arctan(n) - arctan(n-1)] from n = 1 to infinity (where -pi/2 < arctan(n) < pi/2)

If you could give me any tips or advice on how to even start the problem, I would greatly appreciate it.  Thank you.

Answer
Hi Jennifer,

Well, this makes two of us who have never encountered this type of problem before!!

The only thing I found was that if you write out a bunch of terms, you can see what's known as a "collapsing sum":

= [arctan 1 - arctan 0] + [arctan 2 - arctan 1] +

 [arctan 3 - arctan 2] + [arctan 4 - arctan 3] + ...

and as you can see, the arctan 1 's will drop out, the "arctan 2" 's, the "arctan 3" 's , and so on to infinity, so the only term left is the -arctan 0 in the first bracket, and arctan 0 = 0 .

This is about the only thing I can see here.
I hope this is of some help to you.

Steve Holleran