What is the value of $ \sum_{n=1}^\infty  (\tan^{-1}\sqrt{n}-\tan^{-1}\sqrt{n+1})$?

Your answer should be in radians.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
What is the value of $ \sum_{n=1}^\infty  (\tan^{-1}\sqrt{n}-\tan^{-1}\sqrt{n+1})$?

Your answer should be in radians.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
-\frac{\pi}{4}
What is the value of $ \sum_{n=1}^\infty  (\tan^{-1}\sqrt{n}-\tan^{-1}\sqrt{n+1})$?

Your answer should be in radians.
train