Question

The sum of the series \[ \sum_{n=1}^{\infty} \tan^{-1} \left( \frac{2}{n^2} \right) \] is

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{2} \)
• C. \( \pi \)
• D. \( \frac{3\pi}{4} \)

Hint:

For series involving inverse tangent functions, use standard results or approximations for large \( n \) to simplify the sum.

Answer 1

The Correct Option is D

Step 1: Series analysis.
We are asked to evaluate the sum of the series: \[ S = \sum_{n=1}^{\infty} \tan^{-1} \left( \frac{2}{n^2} \right). \] This series involves the inverse tangent function, which is related to well-known series results. Specifically, the general approach to summing such series involves recognizing patterns or using known results for sums of arctangents.
Step 2: Using a known result.
The sum of this series is a known result from advanced series summation techniques, and it is found to converge to \( \frac{3\pi}{4} \), which corresponds to the value for \( S \).
Step 3: Conclusion.
Thus, the correct answer is (D) \( \frac{3\pi}{4} \).