Evaluate the sum of Fib(n)/10^n where Fib(n) denotes the nth Fibonacci number.
That is, calculate 1/10 + 1/10^2 + 2/10^3 + 3/10^4 + 5/10^5 + 8/10^6 + 13/10^7 + ...
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Answer: 10/89
ReplyDeleteS = sum (1 to inf) Fib(n)/10^n
= 1/10 + sum (2 to inf) Fib(n)/10^n
= 1/10 + sum (2 to inf) (Fib(n-1)+Fib(n-2))/10^n
= 1/10 + sum (2 to inf) Fib(n-1)/10^n + sum (2 to inf) Fib(n-2)/10^n
= 1/10 + sum (1 to inf) Fib(n)/10^(n+1) + sum (0 to inf) Fib(n)/10^(n+2)
= 1/10 + S/10 + S/100
Solving S = 1/10 + S/10 + S/100 gives S = 10/89.