Question

Let a_n be Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, ......
a₀ = 1,  a₁ = 1    a₂ = 2    a₃ = 3   ..... 

Find the Sum of the Infinite series: 1 / [ a_n * a_n+2 ] for n = 0 to infinity.

$Sum=\Sigma_{n=0}^{n=\infty}\ \frac{1}{a_n\ a_{n+2}}=\frac{1}{a_0\ a_2}+\frac{1}{a_1\ a_3}+...$

Answer 1

Fibonacci series:  
$a_n+2=a_{n+1}+a_{n}\ \ for\ n=0,1,2,3,....\\\\\frac{1}{a_n}-{\frac{1}{ a_{n+2}}=\frac{a_{n+2}-a_n}{a_n\ a_{n+2}}=\frac{a_{n+1}}{a_n\ a_{n+2}}}\\\\so\ \frac{1}{a_n\ a_{n+2}}}=\frac{1}{a_n\ a_{n+1}}-\frac{1}{a_{n+1}\ a_{n+2}}\\\\ Sum=\Sigma_{n=0}^{n=\infty}\ [ \frac{1}{a_n\ a_{n+1}}-\frac{1}{a_{n+1}\ a_{n+2}} ]\\\\=\frac{1}{a_0a_1}-\frac{1}{a_1a_2}+\frac{1}{a_1a_2}-\frac{1}{a_2a_3}+\frac{1}{a_2a_3}-\frac{1}{a_3a_4}+\frac{1}{a_3a_4}....\infty\\\\=\frac{1}{1*1}\\\\=1$

So the answer is 1.