Inverse Trigonometric Functions
Telescoping Series
Premium Question
Grade 12
Question:
If $\sum_{n=1}^\infty \tan^{-1}\left(\frac{2}{n^2}\right) = \frac{k\pi}{4}$, find $k$.
Step-by-Step Solution
Key Concept: Check if $\tan^{-1}(n + 1) - \tan^{-1}(n - 1) = \tan^{-1}(2/n^2)$. Write out the telescoping pattern and evaluate the limit.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 3