infinity 2 / 2 (R ) | (- x ) J(R) = E | E dx | / RNote that x e^(-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

infinity ----- \ 1 ) ------ / 2 ----- n + 1 n = 1Try for an error less than 10^(-10).

** 3.(a) ** Show that the Fibonacci numbers satisfy an equation
F[n+4] + F[n-4] = c F[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.