Fibonacci Numbers and A Convergent Series.

Hello, Everyone.

From :

Convergent Series .PNG

Need help with proving the above series (green highlight), is a convergent series ?

Thanks in advance for your time and help.

Regards.

Best answer by AlvaroDíaz

Series with therms 1/F.k and 1/F.k+3 converges (see Reciprocal Fibonacci constant - Wikipedia, the free encyclopedia). Also you can decompose S = S1 +S2 with S1 = 1/F.k*1/F.k+2 and S2 = 1/k+1*1/F.k+3. Notice that 1/Fk+1 and 1/F.k+3 are bounded and monotone decrecent sequences. So applying Abbel test (see Abel's test - Wikipedia, the free encyclopedia) to S1 and S2 you prove that S converges because S1 and S2 converges.

Best regards.

Answer

12-Amethyst

Series with therms 1/F.k and 1/F.k+3 converges (see Reciprocal Fibonacci constant - Wikipedia, the free encyclopedia). Also you can decompose S = S1 +S2 with S1 = 1/F.k*1/F.k+2 and S2 = 1/k+1*1/F.k+3. Notice that 1/Fk+1 and 1/F.k+3 are bounded and monotone decrecent sequences. So applying Abbel test (see Abel's test - Wikipedia, the free encyclopedia) to S1 and S2 you prove that S converges because S1 and S2 converges.

Best regards.

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