LeahYoungquist
New member
- Joined
- May 17, 2016
- Messages
- 3
Hello all! I've been trying to figure out this math project for about three weeks now, and everything I try is actually garbage... I'm in Calc III and we're working on Power Series for the most part now. Here is the part of the project I have literally made no headway with!! I've found other examples online going the opposite direction of this one (That's the style of the project. My Prof took famous proofs and switched them around and we have to solve them another way) but it's not been helpful for me...
I think even if I could just get some help with first part then I'd have an idea of how to continue!! Thanks to anyone who take the time even to read this!
We had defined the Fibonacci sequence recursively, but no for-mula was given for its general term. This part of the project will achieve that goal.Consider the function
f(x) =
x
1 − x − x2
1. Suppose that x = sum∞
k=0 akxk. Justify that a1 = 1 and ak = 0 for all k = 1.
2. Show that if f(x) = sum∞
n=1 fnxn, then {fn}∞
n=1 is the Fibonacci sequence.
That is show that f1 = f2 = 1 and fn = fn−1 + fn−2 for n ≥ 3.3. Using partial fractions, find a representation of f(x) in terms of a powerseries.4. Deduce from the steps above a formula for fn for all n ≥ 1
I think even if I could just get some help with first part then I'd have an idea of how to continue!! Thanks to anyone who take the time even to read this!
We had defined the Fibonacci sequence recursively, but no for-mula was given for its general term. This part of the project will achieve that goal.Consider the function
f(x) =
x
1 − x − x2
1. Suppose that x = sum∞
k=0 akxk. Justify that a1 = 1 and ak = 0 for all k = 1.
2. Show that if f(x) = sum∞
n=1 fnxn, then {fn}∞
n=1 is the Fibonacci sequence.
That is show that f1 = f2 = 1 and fn = fn−1 + fn−2 for n ≥ 3.3. Using partial fractions, find a representation of f(x) in terms of a powerseries.4. Deduce from the steps above a formula for fn for all n ≥ 1