Math 210 Sec. 201 Assignment 5 - not for handing in

1.(a) Use repeated integration by parts to find the first 4 terms of an asymptotic series in negative powers of R for

J(R) = e^{R^{2}} ∫_{R}^{∞} e^{−x^{2}} dx

Note that xe^{−x^{2}} has an elementary antiderivative.

(b) Guess a formula for the terms of this series, and prove that this formula is correct. Show that the series never converges.

(c) Estimate the size of the remainders for the partial sums of the series for J(2). When do the terms stop decreasing in magnitude? Which partial sums provide the best approximation to the value of J(2)?

2. Use the Euler-Maclaurin formula to approximate the value of

∑_{n=1}^{∞} \frac{1}{n^{2} + 1}

Try for an error less than 10^{−10}.

3.(a) Show that the Fibonacci numbers satisfy an equation F_{n+4} + F_{n−4} = cF_{n} for some constant c

(and find c).

(b) Using our identity for F_{m+n}, show that F_{kn} is divisible by F_{n} for all positive integers n and k.

(c) Which Fibonacci numbers are divisible by 9? Which are divisible by 3^{k}?

(d) Express 7880997 as a ratio of two Fibonacci numbers.

4. Find a polynomial that approximates ln(1 + x^{3}) on the interval 0 ≤ x ≤ 3 with error less than 10^{−5} at every point.