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).
the -1 in the denominator is outside of the subscript so we actually have \lim_{n->\infty}f(n+1)/(f(n)-1). In this case the -1 is negligible so this has the same limit as f(n+1)/f(n).
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).
The sequence of arguments is the following. 1) We know from the Lemma that the sequence 'H_n - ln(n) - gamma' converges to '0'. 2) Hence the subsequences 'H_{f_{n+1} - ln(f_{n+1} - gamma' and 'H_{f_n - 1} - ln(f_n - 1) - gamma' converge as well to 0. 3) Define two new sequences 'a_n = H_{f_{n+1} - ln(f_{n+1} - gamma' and 'b_n = H_{f_n - 1} - ln(f_n - 1) - gamma' and we know from #2 that they both converge to 0. 4) Hence from 'Algebraic Limit Theorem', their difference also converges to '0'. 5) We know separately that 'ln(f_{n+1}) - ln(f_n - 1)' is another convergent sequence which converges to 'ln(Phi)'. 6) Hence the sum of 'a_n - b_n', which converges to 0, and 'ln(f_{n+1}) - ln(f_n - 1)', which converges to 'ln(Phi)', must also converge because it is a sum of two convergent sequences and 'Algebraic Limit Theorem' states that sums and differences of convergent sequences converge. 7) Finally, you get, in one shot, that H_{f_{n+1}}-H_{f_n-1} is convergent, as well as that it converges to ln(Phi).
Best maths channel by far.
Try to solve this problem th-cam.com/video/mM4C5OnNIi8/w-d-xo.html
Not by far it is and it will be.
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).
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
timecode 15:12 : lim n--> infini (f(n+1)/f(n-1))= 2.618 (= 1 + phi)
the -1 in the denominator is outside of the subscript so we actually have \lim_{n->\infty}f(n+1)/(f(n)-1). In this case the -1 is negligible so this has the same limit as f(n+1)/f(n).
Shouldn't be there f(n+1)/f(n-1) instead of f(n+1)/(f(n)-1)?
@@miro.s No. The lower bound is f(n)-1.
I also thought the -1 was in the subscript and was sinilarly confused, thanks for asking that question Stewart
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.
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).
Very interesting video. I understood everything! Thanks!
Does anyone see why H_{f_{n+1}}-H_{f_n-1} is convergent?
The sequence of arguments is the following. 1) We know from the Lemma that the sequence 'H_n - ln(n) - gamma' converges to '0'. 2) Hence the subsequences 'H_{f_{n+1} - ln(f_{n+1} - gamma' and 'H_{f_n - 1} - ln(f_n - 1) - gamma' converge as well to 0. 3) Define two new sequences 'a_n = H_{f_{n+1} - ln(f_{n+1} - gamma' and 'b_n = H_{f_n - 1} - ln(f_n - 1) - gamma' and we know from #2 that they both converge to 0. 4) Hence from 'Algebraic Limit Theorem', their difference also converges to '0'. 5) We know separately that 'ln(f_{n+1}) - ln(f_n - 1)' is another convergent sequence which converges to 'ln(Phi)'. 6) Hence the sum of 'a_n - b_n', which converges to 0, and 'ln(f_{n+1}) - ln(f_n - 1)', which converges to 'ln(Phi)', must also converge because it is a sum of two convergent sequences and 'Algebraic Limit Theorem' states that sums and differences of convergent sequences converge. 7) Finally, you get, in one shot, that H_{f_{n+1}}-H_{f_n-1} is convergent, as well as that it converges to ln(Phi).
This is so true I can understand what your saying and real understand your point. Thank you