A Fibonacci bounded partial sum of the Harmonic series.

We determine the limit of a certain sequence defined in terms of Fibonacci and Harmonic numbers.

[joshhickman]

The great part is, the proof is super general, proves similar results for all sequences, where the limit is the ln of the ratio of bounds (taken in the limit, of course).

[hybmnzz]

Showing the Euler-Mascheroni constant exists as a casual lemma was pretty exciting.

[probabilitystochasticproce]

The way you have drawn the diagram, the integral upper limit at t=1:51 should be (n+1), not n. This gives = ln(1+ 1/n) > 0.

[hamsterdam]

so you can apply (almost) last formula to any geometrically increasing sequence where limit (a_(n+1)/(a_n-1)) converges to a positive number

[dugong]

I got lost in your proof that the sequence is decreasing, but before watching I figured in the limit the sum is the same as the integral (and I think it could be shown that it's squeezed between 2 integrals that are equal in the limit, one using those limits and one using limits of Fn+1 and F(n+1)+1) of 1/x from Fn to F(n+1) and that in the limit Fn=phi^n/sqrt(5) so the integral is ln(phi^(n+1)/sqrt(5)) - ln(phi^(n)/sqrt(5)) which is ln(phi).

[stewartcopeland]

timecode 15:12 : lim n--> infini (f(n+1)/f(n-1))= 2.618 (= 1 + phi)

[NoName-ehfz]

Very interesting video. I understood everything! Thanks!

[OrlandoRiveraLetelier]

Does anyone see why H_{f_{n+1}}-H_{f_n-1} is convergent?

[georgettebeulah]

This is so true I can understand what your saying and real understand your point. Thank you