An interesting question about series

Art of Problem Solving

AoPS Online

College Math Topics in undergraduate and graduate studies

Topics in undergraduate and graduate studies

3 M G

BBookmark VNew Topic kLocked

College Math Topics in undergraduate and graduate studies

Topics in undergraduate and graduate studies

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

An interesting question about series

G H J

Reveal topic tags

L

G H BBookmark kLocked kLocked NReply

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

10 posts

#1 h Mar 23, 2025, 7:39 PM

Y by

Calculate $\sum_{n=1}^{+\infty} \frac{(-1)^n}{F_n F_{n+2}}$ where $(F_n)$ denotes the Fibonacci sequence.**

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

10 posts

#2 h Mar 23, 2025, 8:17 PM

Y by

Is that correct: $\sum_{n=1}^{+\infty} \frac{(-1)^n}{F_n F_{n+2}}$
$F_{n+2} = F_{n+1} + F_n \implies \frac{F_{n+2}}{F_n} = \frac{F_{n+1}}{F_n} + 1$
$\lim_{n \to \infty} \frac{F_{n+1}}{F_n} = \varphi = \frac{1+\sqrt{5}}{2}$
$\frac{F_{n+2}}{F_n} > \frac{F_{n+1}}{F_n} > 1 \implies \frac{1}{F_n F_{n+2}} < \frac{1}{F_n^2}$
$\left|\frac{(-1)^n}{F_n F_{n+2}}\right| < \frac{1}{F_n^2} < \frac{1}{n^2}$
$\implies \sum_{n=1}^{+\infty} \frac{(-1)^n}{F_n F_{n+2}} \text{ converges absolutely}$
$F_{n+1}F_{n-1} - F_n^2 = (-1)^{n-1}$
$F_{n+2}F_n - F_{n+1}^2 = (-1)^n$
$\frac{1}{F_n F_{n+2}} = \frac{F_{n+1}^2 - (-1)^n}{F_n F_{n+1}^2} = \frac{F_{n+1}}{F_n} \cdot \frac{1}{F_{n+1}^2} - \frac{(-1)^n}{F_n F_{n+1}^2}$
$\sum_{n=1}^{+\infty} \frac{(-1)^n}{F_n F_{n+2}} = \sum_{n=1}^{+\infty} (-1)^n \left[\frac{F_{n+1}}{F_n} \cdot \frac{1}{F_{n+1}^2} - \frac{(-1)^n}{F_n F_{n+1}^2}\right]$
$= \sum_{n=1}^{+\infty} \frac{(-1)^n F_{n+1}}{F_n F_{n+1}^2} - \sum_{n=1}^{+\infty} \frac{1}{F_n F_{n+1}^2}$
$= \sum_{n=1}^{+\infty} \frac{(-1)^n}{F_n F_{n+1}} - \sum_{n=1}^{+\infty} \frac{1}{F_n F_{n+1}^2}$
$= -\frac{1}{2} - 0 = -\frac{1}{2}$

This post has been edited 1 time. Last edited by Ayoubgg, Mar 23, 2025, 8:19 PM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

611 posts

#3 h Yesterday at 1:56 PM

Y by

Your proof of absolute convergence is correct, but I can't follow the last four lines. It seems there are some algebraic mistakes and the result seems to be wrong.

The sum is a special case of the infinite sums considered in the article 'SUMMATION OF INFINITE FIBONACCI SERIES' by 'ALFRED BROUSSEAU', see (3) in

https://www.fq.math.ca/Scanned/7-2/brousseau1.pdf

The proof in the given reference is based on the identity $F_{n+2}F_{n-1} - F_{n+1} F_n = (-1)^n$ and a resulting telescoping sum, so this seems to be similar to what you did.

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a