Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>Let Un =
n!
(n+2)! where n ∈N. If Sn = Pn
r=1 Ur, then limn→∞Sn equals:</p>
Step-by-Step Solution
Key Concept: Simplify the factorial term to create a telescoping sum.
<p><strong>1</strong>: Un =</p> n! (n+2)! = n! (n+2)(n+1)n! = 1 (n+1)(n+2).<p><strong>2</strong>: Using partial fractions: Un =</p> 1 n+1 - 1 n+2.<p><strong>3</strong>: Sum Sn =</p> 1 2 -1 3 + 1 3 -1 4 + \cdot \cdot \cdot + 1 n+1 - 1 n+2 .<p><strong>4</strong>: Sn = 1</p> 2 - 1 n+2 n\to \infty ---\to 1/2.
Correct Answer: 3