If `tan^(-1)(1/(1+1. 2))+tan^(-1)(1/(1+2. 3))+...+tan^(-1)(1/(1+n(n+1)))=tan^(-1)theta,` then find the value of `thetadot`

Explanation

To solve the given problem, we need to evaluate the expression:

\[ \tan^{-1}\left(\frac{1}{1 + 1 \cdot 2}\right) + \tan^{-1}\left(\frac{1}{1 + 2 \cdot 3}\right) + \ldots + \tan^{-1}\left(\frac{1}{1 + n(n + 1)}\right) = \tan^{-1}(\theta) \]

### Step 1: Simplifying the Terms
Each term in the series can be rewritten using the identity:
...