<p>lim
n→∞tan
n
X
r=1
tan−1
1
1 + r + r2
!
is equal to:</p>
Step-by-Step Solution
Key Concept: Convert each term into a telescoping difference of inverse tangents.
<p>Use</p> tan-1 1 1 + r + r2 = tan-1 1 r -tan-1 1 r + 1 , because tan(A -B) = tan A -tan B 1 + tan A tan B . Therefore, n X r=1 tan-1 1 1 + r + r2 = tan-1(1) -tan-1 1 n + 1 . Now take tangent: tan tan-1(1) -tan-1 1 n + 1 = 1 - 1 n+1 1 + 1 n+1 = n n + 2. Hence, lim n\to \infty n n + 2 = 1. Shortcut / Fast View Whenever you see 1 r2+r+1 inside tan-1, try tan-1 1 r -tan-1 1 r+1 .
Correct Answer: (1)