Assignment 3 Due Thursday, March 9thAs always, show your work and explain your answer.1. Consider the linear recurrence a0 = 1, a1 = 0, and for i ≥ 2, ai = 5ai−1 − 6ai−2.(a) Let f(x)...

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Answered Same DayMar 09, 2023

Answer To: Assignment 3 Due Thursday, March 9thAs always, show your work and explain your answer.1....

Aditi answered on Mar 09 2023
54 Votes
SOL
1.
(a) We start by multiplying the recurrence by xi and summing from i = 2 to infinity:
ai xi = 5ai-1 xi - 6ai-2 xi ∑ ai xi = ∑ 5ai-1 xi - ∑ 6ai-2 xi ∑ ai xi = 5x(∑ ai-1 xi) - 6x^2(∑ ai-2 xi)
We can rewrite the last equation as:
f(x) - a0 - a1x = 5x(f(x) - a0) - 6x^2f(x)
Sim
plifying and solving for f(x), we get:
f(x) = (1 - 5x)/(1 - 5x + 6x^2)
(b) To expand f(x) using partial fractions, we need to factor the denominator:
1 - 5x + 6x^2 = (2x - 1)(3x - 1)
We can then write f(x) as:
f(x) = A/(2x - 1) + B/(3x - 1)
Multiplying both sides by (2x - 1)(3x - 1), we get:
1 - 5x + 6x^2 = A(3x - 1) + B(2x - 1)
Substituting x = 1/2 and x = 1/3, we can solve for A and B:
A = 4, B = -3
Therefore, we can write f(x) as:
f(x) = 4/(2x - 1) - 3/(3x - 1)
To determine the coefficient an, we need to expand f(x) in a power series:
f(x) = 4/(2x - 1) - 3/(3x - 1) = 4/2 ∑ (x/2)^n - 3/3 ∑ (x/3)^n = 2 ∑ (x/2)^n - ∑ (x/3)^n
Using the formula for the coefficient of xn in a power series, we can see that:
an = 2^(n+1) - 3^n
for all n ≥ 0.
2.
(a) To express f(x) as a sum of two terms, we need to factor the denominator first:
x^2 - x + 3 = (x - (1+2i))(x - (1-2i))
Then we can write:
f(x) = A/(x - (1+2i)) + B/(x - (1-2i))
where A and B are constants to be determined. We can find A and B by multiplying both sides by the denominators and simplifying:
1 = A(x - (1-2i)) + B(x - (1+2i)) 1 = (A+B)x - (A(1-2i) + B(1+2i))
Equating the coefficients of x and the constant term, we get:
A + B = 0 A(1-2i) + B(1+2i) = 1
Solving for A and B, we get:
A = 1/4 + i/4 B = 1/4 - i/4
Therefore, we can express f(x) as:
f(x) = (1/4 + i/4)/(x - (1+2i)) + (1/4 - i/4)/(x - (1-2i))
(b) To determine the coefficient ar, we can use the formula:
ar = 1/(2πi) ∮ γ f(x)/xr+1 dx
where γ is a closed contour in the complex plane that encloses the origin in a counterclockwise direction, and the integral is taken along γ.
Using the residue theorem, we can evaluate this integral as the sum of the residues of f(x)/xr+1 at the poles inside γ. In this case, the poles are at x = 1+2i and x = 1-2i. We can compute the residues at these poles as:
Res(f(x)/xr+1, x = 1+2i) = lim(x→1+2i) ((x - (1+2i))f(x))/xr+1 = (1/4 + i/4)/(r+1)(1-2i)r
Res(f(x)/xr+1, x = 1-2i) = lim(x→1-2i) ((x - (1-2i))f(x))/xr+1 = (1/4 - i/4)/(r+1)(1+2i)r
Therefore, the coefficient ar is given by:
ar = 1/(2πi) ∮ γ f(x)/xr+1 dx = Res(f(x)/xr+1, x = 1+2i) + Res(f(x)/xr+1, x = 1-2i) = (1/4 + i/4)/(r+1)(1-2i)r + (1/4 - i/4)/(r+1)(1+2i)r
Simplifying this expression, we get:
ar = (1/2r+2) ((-2/5)^(r+2) + 3(2/3)^(r+2))
3.
(a) We can express each odd integer as 2j+1 for some non-negative integer j. Then the equation becomes b1 + b2 + · · · + bk = 2j1 + 2j2 + · · · + 2jk, where each bi is a non-negative integer and each j1, j2, . . . , jk is a non-negative integer. Now, we can...
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