Evened Fibonacci sum

Calculus Level 4

Evaluate $\sum_{n=0}^{\infty}(-1)^n\frac{F_{2n}}{9^n}$

Notation: $F_n$ denotes the $n^\text{th}$ Fibonacci number .

The answer is -0.0825688.

2 solutions

Naren Bhandari
Oct 24, 2020

Since the generating function of Fibonacci series is $\mathcal{F}_n(x)= \sum_{n\geq 0} F_n x^n=\frac{x}{1-x-x^2}, \;\;x<\phi^{-1}$ where $\phi$ is golden ratio and since it is easy to derive by it recursive relation so it is easy to note that $\displaystyle \mathcal{F}_{2n}(x)$ $\sum_{n\geq 0} F_{2n} x^{2n}=\frac{1}{2}\left(\mathcal{F}_n(x)+\mathcal{F}_n(-x)\right)$ giving us the last expression for $\mathcal{F}_{2n}(x)=\frac{1}{2}\left(\frac{x}{1-x-x^2}-\frac{x}{1+x-x^2}\right)=\frac{x^2}{x^4-3x^2+1}$ now replacing $x$ by $\sqrt{x}$ and them $x=-\frac{1}{9}$ and simplification gives us $\sum_{n\geq 0} (-1)^n\frac{F_{2n}}{9^n}=\frac{x}{x^2-3x+1}\Bigg|_{x=-\frac{1}{9}}=-\frac{9}{109}$

We can note that, for all $x<\phi^{-1}$ $\sum_{n\geq 0} F_{2n} x^n=\frac{x}{x^2-3x+1}$ and $\sum_{n\geq 0} F_{2n+1} x^n=\frac{1-x}{x^2-3x+1}$

Another great question. +1+1+1

I recommend that you change your question to...

... This series is equal to $-\frac ab$ , where $a$ and $b$ are coprime positive integers. What is $a+b$ ?

With an answer of $9 + 109 = \boxed{118}$ .

Also, is this FB group any good? I might need to get an FB account for this.

Pi Han Goh - 7 months, 2 weeks ago

Thank you sir.

Firstly I thought like of your recommendation however, I thought to share the same problem as it is the linked one. So I didn't make any changes.

Yes, it is a facebook group of Romanian mathematical Magazine which share problem on facebook.

Naren Bhandari - 7 months, 2 weeks ago

I entered in the right answer but for some reason Brilliant didn't accept the negative sign. I figured I had done something incorrectly but I guess not...

Alexander McDowell - 7 months, 2 weeks ago

@Brilliant Mathematics @Silas Hundt

Pi Han Goh - 7 months, 2 weeks ago

Since I can't give a solution because of an error with Brilliant (even though my answer was correct), here is my solution:

The fibonacci numbers can be generated using the function F(n) = (phi1^n - phi2^n)/sqrt(5) where phi1 = (1 + sqrt(5)) / 2 and phi2 = (1 - sqrt(5)) / 2. Thus, the given summation is equivalent to (sum((-phi1^2 / 9)^n) - sum((-phi2^2 / 9)^n))/sqrt(5). Each of the sums is an infinite geometric series with r < 1 so we can use the formula 1/1-r to calculate the infinite sum. The sum therefore equals (1/(1 + phi1^2/9) + 1/(1 + phi2^2/9))/sqrt(5). I would show all the steps I took to get the final answer but it would make the comment look very messy. Through a lot of manipulation, you should get -9/109 as the final answer.

Alexander McDowell - 7 months, 2 weeks ago

Sorry, I cannot be helpful in case of technical issues. I hope @Brilliant Mathematics can help here.

Naren Bhandari - 7 months, 2 weeks ago

What is $\phi^{-1}$ ?

Chew-Seong Cheong - 7 months, 2 weeks ago

reciprocal of the golden ratio. Yeah, the solution should have mentioned it.

Pi Han Goh - 7 months, 2 weeks ago

Thanks. Yes, the solution should have mentioned it.

Chew-Seong Cheong - 7 months, 2 weeks ago

I have mentioned about it. Please check it.

Naren Bhandari - 7 months, 2 weeks ago

Sorry, I didn't check

Chew-Seong Cheong - 7 months, 2 weeks ago

Arindam Ghosh
Oct 24, 2020

I know this is the cheating way but ...

I can see the sum is converging to a value -0.0825688

def fibo(n):
    if n < 1:
        return 0
    elif n == 1:
        return 1
    else:
        return fibo(n-1) + fibo(n-2)

sum = 0
for i in range(1,100):
    sum += ((-1/9)**i)*fibo(2*i)

    print(sum)

You could greatly reduce the runtime if you add the following two lines at the top:

from functools import lru_cache
@lru_cache(maxsize = 100)
def fibo(n):
    if n < 1:
        return 0
    elif n == 1:
        return 1
    else:
        return fibo(n-1) + fibo(n-2)

sum = 0
for i in range(1,100):
    sum += ((-1/9)**i)*fibo(2*i)

    print(sum)

For more information, check out this video .

Pi Han Goh - 7 months, 2 weeks ago

Evened Fibonacci sum

The answer is -0.0825688.

2 solutions

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