Question

The sum of the infinite series $ \cot^{-1} \left( \frac{7}{4} \right) + \cot^{-1} \left( \frac{19}{4} \right) + \cot^{-1} \left( \frac{39}{4} \right) + \cot^{-1} \left( \frac{67}{4} \right) + ... $ is:

• A. \( \frac{\pi}{2} + \tan^{-1} \left( \frac{1}{2} \right) \)
• B. \( \frac{\pi}{2} - \cot^{-1} \left( \frac{1}{2} \right) \)
• C. \( \frac{\pi}{2} + \cot^{-1} \left( \frac{1}{2} \right) \)
• D. \( \frac{\pi}{2} - \tan^{-1} \left( \frac{1}{2} \right) \)

Hint:

When dealing with infinite series involving cotangent or tangent, look for cancellation patterns and simplify the terms.

Answer 1

The Correct Option is D

Let the sum of the series be \( S \), where the general term is \( T_n \): \[ T_n = \cot^{-1} \left( \frac{4n}{2n^2 + 3} \right) \] This can be simplified as: \[ T_n = \cot^{-1} \left( \frac{n + \frac{1}{2}}{1 + \left( n + \frac{1}{2} \right)^2} \right) \] Now the series becomes: \[ S = T_1 + T_2 + \dots = \cot^{-1} \left( n + \frac{1}{2} \right) - \cot^{-1} \left( n - \frac{1}{2} \right) \]
Therefore, the sum of the infinite series is: \[ S = \frac{\pi}{2} - \tan^{-1} \left( \frac{1}{2} \right) \]
Thus, the correct answer is \( \frac{\pi}{2} - \tan^{-1} \left( \frac{1}{2} \right) \).

Answer 2

Step 1: Understanding the series. The given series is: \[ \cot^{-1} \left( \frac{7}{4} \right) + \cot^{-1} \left( \frac{19}{4} \right) + \cot^{-1} \left( \frac{39}{4} \right) + \cot^{-1} \left( \frac{67}{4} \right) + \cdots \] This is a standard arccotangent series.

Step 2: Identifying the pattern. The series consists of terms of the form: \[ \cot^{-1} \left( \frac{4n + 3}{4} \right), \quad n = 1, 2, 3, \dots \] By using the identity for the sum of arccotangents: \[ \cot^{-1}(x) + \cot^{-1}(y) = \cot^{-1} \left( \frac{xy - 1}{x + y} \right) \] we can simplify the series.

Step 3: Applying the formula to the series. By applying the identity iteratively and simplifying, we can find that the sum of the infinite series converges to: \[ \pi - \cot^{-1} \left( \frac{1}{2} \right) \]

Step 4: Conclusion. The sum of the infinite series is: \[ \boxed{\pi - \tan^{-1} \left( \frac{1}{2} \right)} \] Final Answer: \[ \boxed{4}. \]