Let's break down and analyze the problem step by step to arrive at the solution clearly.

Step 1: Understanding the expression

We are given: $\tan^{-1}\left(\frac{1 + r + r^2}{1}\right) = \tan^{-1}\left(\frac{1 + (r+1)r}{(r+1)-r}\right) = \tan^{-1}(r+1) - \tan^{-1}(r)$ This shows that the given expression can be simplified to the difference of two arctangents: $\tan^{-1}(r+1) - \tan^{-1}(r)$

Step 2: Summation

The next step is to apply this to the summation: $\sum_{r=1}^{n} \tan^{-1}\left(\frac{1 + r + r^2}{1}\right)$ This becomes: $\sum_{r=1}^{n} \left(\tan^{-1}(r+1) - \tan^{-1}(r)\right)$ This is a telescoping series, where most terms cancel out. Expanding the series: $\left(\tan^{-1}(2) - \tan^{-1}(1)\right) + \left(\tan^{-1}(3) - \tan^{-1}(2)\right) + \cdots + \left(\tan^{-1}(n+1) - \tan^{-1}(n)\right)$ All the intermediate terms cancel, leaving: $\tan^{-1}(n+1) - \tan^{-1}(1)$

Step 3: Taking the limit as $n \to \infty$

Now, consider the limit as $n \to \infty$: $\lim_{n \to \infty} \left(\tan^{-1}(n+1) - \tan^{-1}(1)\right)$ As $n \to \infty$, we know that: $\tan^{-1}(n+1) \to \frac{\pi}{2}$ Thus: $\lim_{n \to \infty} \left(\frac{\pi}{2} - \tan^{-1}(1)\right) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}$

Step 4: Final expression

The final part of the problem is: $\lim_{n \to \infty} \tan\left(\tan^{-1}(n+1) - \tan^{-1}(1)\right)$ We already calculated that: $\tan^{-1}(n+1) - \tan^{-1}(1) \to \frac{\pi}{4}$ Thus: $\tan\left(\frac{\pi}{4}\right) = 1$

Conclusion

Therefore, the result of the given expression is: $\boxed{1}$

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Do you need more details or have any questions? Here are some related questions you might want to explore:

1. What is the behavior of the arctangent function as $r \to \infty$?
2. How can we identify and simplify a telescoping series?
3. How does the limit of arctangent functions behave?
4. Why does $\tan\left(\frac{\pi}{4}\right) = 1$?
5. How can we generalize this result for other similar functions?

Tip: Telescoping series are very useful in simplifying large sums, especially when terms cancel out step by step.