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#### Pranav

##### Well-known member

- Nov 4, 2013

- 428

**Problem:**

Evaluate

$$\lim_{n\rightarrow \infty} \left(\sum_{r=1}^n (\arctan(2r^2))-\frac{n\pi}{2}\right)$$

**Attempt:**

I tried evaluating the summation but couldn't. Had the problem involved $\arctan(1/(2r^2))$, I could rewrite it as

$$\arctan\left(\frac{2r+1-(2r-1)}{1+(2r+1)(2r-1)}\right)$$

and evaluating the sum would be quite easy but honestly, I have no idea for the given problem.

Any help is appreciated. Thanks!