Evaluate 6/( (9 - 4)(3 - 2) ) + 36/( (27 - 8)(9 - 4) ) + ... + 6n/( (3n+1 - 2n+1)(3n - 2n) ) + ... .
Solution
Answer: 2.
An easy induction on n shows that the sum of the first n terms is 6(3n - 2n)/(3n+1 - 2n+1). We may write this as 2 (1 - (2/3)n)/(1 - (2/3)n+1), which shows that it tends to 2 as n tends to infinity.
© John Scholes
jscholes@kalva.demon.co.uk
7 Jan 2001