In today’s lecture I left two computations for you to carry out. Anyone? (If you know how to do both, please show us one, and let someone else do the other one.)

1. Compute the constants such that .

2.Today I proved this formula for the Fibonacci numbers: Derive this formula directly from the generating function .

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Problem 1:

latex \frac{A}{1- \alpha x} + \frac{B}{1- \beta x}$ =

Now, we make equal the coefficients:

(a)

(b)

(c)

(d) $- \alpha \beta =1$

(c) and (d) implies that , so and .

so if we take we get

(a) and (b) implies that

Notice that , so and because of (a)

Andrés F. Higuera

andresfelipehiguera.wix.com/economia

University of the Andes

Department of Economics

Problem 1:

latex \frac{A}{1- \alpha x}$ + =

Now, we make equal the coefficients:

(a)

(b)

(c)

(d) $- \alpha \beta =1$

(c) and (d) implies that , so = and .

so if we take = we get

(a) and (b) implies that

Notice that , so and because of (a)

Andrés F. Higuera

andresfelipehiguera.wix.com/economia

University of the Andes

Department of Economics

Well, I was able to work out the first one, computing the constants , but I made a bit of an assumption I’m not sure was okay.

Oops that comment wasn’t complete. Here goes:

We have . I combined the fractions on the right to get . So then I assumed that the denominators needed to be the same, meaning and . I solved for in the first equation, and substituted it into the second, giving . Then I solved for and got . So then I substituted this answer back into and got .

Now I went back to my original equation, and got rid of my denominators be multiplying both sides by each denominator. So I got .

(I think I wrote that correct…) So from this we know that coefficients have to be the same, and in particular, I used that and . Since , I got and then substituted in and got that .

I would love input on this! Thanks.

Your assumption is correct, that’s what partial fractions is about. However, the solution of Is .

In particular Federico took .That’s the reason why your expressions have a diferent sing in comparison to federico’s solution.

Yes, I just chose differently and realized that’s which my signs where switched for my and pairs. Thanks!

Lemme take a stab at the second question. The series can be expanded as

I think that Federico meant this expression for the nth Fibonacci

number:

To obtain this from the series, consider the coefficient of the term

when is fully expanded. The last term

from the partial expansion shown above that contributes to this coefficient

is (why?), which by using binomial

expansion we see becomes

The only part of this which contributes to the coefficient of

is the first term on the inside of the parentheses multiplied by the

on the outside, or

Combinatorially, we can think of the binomial coefficients as the

number of ways to choose a certain number of left terms and right

terms from each of the factors. That is, since

then the number of ways of choosing left terms () and

right terms () to get is

or equally

where the number on the top is the total number of factors.

Okay, so we’ve established the first term of : .

Where does the second term come from? It comes

from the coefficient of when

is expanded fully. In this case we have factors to choose terms

from, and 1 of those terms must be a term. Thus, the contribution

is .

Continuing in this fashion, we see that indeed