Power Series Math Project

[LeahYoungquist]

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

 

[Deleted member 4993]

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

Can you factorize:

g(x) = x² + x - 1

 

[stapel]

We had defined the Fibonacci sequence recursively, but no formula was given for its general term. This part of the project will achieve that goal.

Consider the function \(\displaystyle \, f(x)\, =\, \dfrac{x}{1\, -\, x\, -\, x^2}\)

Have I corrected guessed the function above? If not, please reply with clarification.

1. Suppose that x = sum∞

k=0 akxk.

I'm sorry, but I don't know what this means. I suspect that you're doing some sort of summation, but it looks like your equation is as follows:

. . . . .\(\displaystyle \displaystyle x\, =\, \sum_{k\, =\, 0}^{\infty}\, akxk\)

...which makes no sense. Please reply with clarification. (You can use standard web-safe math formatting, as explained here.) Thank you!

 

[LeahYoungquist]

Have I corrected guessed the function above? If not, please reply with clarification.


I'm sorry, but I don't know what this means. I suspect that you're doing some sort of summation, but it looks like your equation is as follows:

. . . . .\(\displaystyle \displaystyle x\, =\, \sum_{k\, =\, 0}^{\infty}\, akxk\)

...which makes no sense. Please reply with clarification. (You can use standard web-safe math formatting, as explained here.) Thank you!

Oh good lord this did not post correctly at all!! I'll repost it after class (in an hour or so)!! My bad yikes x)

 

[LeahYoungquist]

Power Series Math Project (second attempt)

Consider the function f(x)=x/(1-x-x^2)
1. Suppose that x= Sum k=0 to infinity (a_k * x^k). Justify that a_1=1 and a_k=0 for all k =/=1.

2. Show that if f(x)= Sum n=1 to infinity (f_n * x^n), then {fn} from n=1 to infinity is the Fibonacci sequence.

3. Using partial fractions, find a representation of f(x)in terms of a power series.

4. Deduce from the steps above a formula for f_n for all n >=1.

 

[stapel]

The following is my interpretation:

\(\displaystyle \mbox{Consider the function }\, f(x)\, =\, \dfrac{x}{1\, -\, x\, -\, x^2}\)

\(\displaystyle \mbox{1. Suppose that }\, x\, =\, \)\(\displaystyle \displaystyle \sum_{k\, =\, 0}^{\infty}\, \left(\, a_k\,\cdot\, x^k\, \right).\)

\(\displaystyle \mbox{Justify that }\, a_1\, =\, 1\, \mbox{ and }\, a_k\, =\, 0\, \mbox{ for all }\, k\, \neq\, 1.\)

\(\displaystyle \mbox{2. Show that if }\, f(x)\, =\, \)\(\displaystyle \displaystyle \sum_{n\, =\, 1}^{\infty}\, \left(\, f_n\, \cdot \, x^n\, \right),\, \)

\(\displaystyle \mbox{ then }\, \{f_n\}\, \mbox{ from }\, n\, =\, 1\, \mbox{ to }\, \infty\, \mbox{ is the Fibonacci sequence.}\)

\(\displaystyle \mbox{3. Using partial fractions, find a representation of }\, f(x)\, \mbox{ in terms of a power series.}\)

\(\displaystyle \mbox{4. Deduce from the steps above a formula for }\, f_n\, \mbox{ for all }\, n\, \geq\, 1.\)

Is this correct? Also, what is the definition of "f_n" and of "{f_n}"?

How far have you gotten in attempting this exercise?

Please be complete. Thank you!