Compute \(\sum_{n = 1}^\infty \frac{1}{F_n F_{n + 2}}\)
where \(F_n\) denotes the nth Fibonacci number, so \(F_0 = 0\) and \(F_1 = 1\)
Just sum up about 20 terms and you will quickly see that it converges to 1:
\sum_{n = 1}^\infty \frac{1}{F_n F_{n + 2}}
0.5
0.3333333333
0.1
0.0416666666 7
0.0153846153 8
0.0059523809 52
0.0022624434 39
0.0008658008 658
0.0003304692 664
0.0001262626 263
0.0000482229 8307
0.0000184202 7704
0.0000070358 12285
0.0000026874 56833
0.0000010265 14879
0.0000003920 941277
0.0000001497 665813
0.0000000572 0575078
0.0000000218 5065141
0.0000000083 46206313
Sum Total = 0.9999999948.........
