Working with the Fibonacci sequence

Calculus Level 4

$\large f(x) = \sum_{n=0}^{\infty} F_{n}x^{n}$

Define $f(x)$ as the series above with $F_{n}$ as the Fibonacci sequence with $F_{0} = 0, F_{1}=1$ . Calculate $f(\frac{1}{2})$ !

If you think the answer diverges, submit your answer as -1000.

Bonus : Can you generalize the formula $\large \displaystyle f(x) = \sum_{n=0}^{\infty} a_{n}x^{n}$ , which the sequence $a_{n+2} = a_{n+1} + a_{n}, a_0 = p, a_1 = q$ ? And what is the limit/boundaries of the generalization?

The answer is 2.

1 solution

Pritish Reddy
May 30, 2015

Let us define $f(x)_{(p,q)}$ as the summation with $a_{0}=p$ and $a_{1}=q$ We can then express $f(x)_{(p,q)}= p+(px+q)(f(x)_{(0,1)})$ We require $f(x)_{(0,1)}$ in this question, Let us consider $f(x)_{(1,1)}$ , We observe that the value of this sequence is the same as $f(x)_{(0,1)}$ since the first term of the sequence, which makes the only difference between the two sequences, is zero. $f(x)_{(1,1)}=f(x)_{(0,1)}$ Substituting this result in the earlier equation, $f(x)_{(1,1)}=1+(x+1)(f(x)_{(0,1)})$ $f(x)_{(0,1)}=1+(x+1)(f(x)_{(0,1)})$ $f(x)_{(0,1)}=\frac{1}{x}$ Therefore, $f(1/2)_{(0,1)}=\boxed{2}$

Uh, you forgot to factorial it.

Stewart Gordon - 6 years ago

Thank you for pointing it out Stewart, but i believe it was only the excitement of the question poster and not really the factorial that was asked for since the function clearly does not always take integer values, I am cynical about how much sense it makes to ask for the factorial value of a real value function.

Pritish Reddy - 6 years ago

Hmm. I suppose what it boils down to is that by being told to take the factorial, you are basically being told that the result happens to be an integer. In a fractional expression, often the denominator isn't a nonzero-valued function, but the value happens to be nonzero and so the expression is considered valid. So I suppose the same would apply here.

In this instance, luckily 2! = 2 and so it doesn't make a difference. But if it did, then the problem setter would have to be careful not to include a punctuation mark where it can validly be interpreted as a mathematical symbol with a totally different meaning.

Stewart Gordon - 6 years ago

@Stewart Gordon – I understand your view about the fractional expression, but I would like to argue that the factorial operator works only in the domain of positive integers and the function at hand here is a real value function which only happens to take a real value that is also an integer. That is why I challenge the notion of finding the factorial. Anyway, we will just have to live with the happy coincidence of 2!=2 unless the poster chooses to comment.

Pritish Reddy - 6 years ago

Please look at the last but one step. Will it be ( -1/x) or (1/x) ?Pl clarify.. Thanks for splendid solution.

Prabir Chaudhuri - 5 years, 11 months ago

Working with the Fibonacci sequence

The answer is 2.

1 solution

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