1 / | 1/2 | x dx | / 0with ME(n), TE(n) and SE(n) the errors for Midpoint, Trapezoid and Simpson's Rules. Plot n^(3/2) ME(n), n^(3/2) TE(n), and n^(3/2) SE(n) (for even n) on the same graph. What can you conclude?

**(b) **
Approximately what n would be needed to make the Simpson's Rule
error be less than 10^(-9)?

**(c) **
Transform

infinity / | 1 | --------------- dx | 3 1/2 / x + (x + 1) 0into a proper integral with a smooth integrand, and use Simpson's Rule to find an approximation with error less than 10^(-6).

**2.(a) ** Catalan's constant is defined as

infinity ----- i \ (-1) C = ) ---------- / 2 ----- (2 i + 1) i = 0By finding bounds on the tail of this series, determine the value of C with an error less than 10^(-8). Compare to Maple's "Catalan". Hint: consider the terms in pairs:

infinity ----- \ C = ) (a + a ) / 2j 2j+1 ----- j = 0

/ n \ |----- | | \ | gamma = limit | ) 1/i - ln(n)| n -> infinity | / | |----- | \i = 1 /By finding bounds on the tail of the convergent series

infinity ----- \ ) (1/i + ln(i) - ln(i + 1)) / ----- i = 1determine the value of gamma with an error less than 10^(-8). Compare to Maple's "gamma".

**3.(a) ** Let

2 Pi / 1 | f(x) = ---- | exp(x cos(t)) dt 2 Pi | / 0Maple can't evaluate this integral, but in fact it turns out to be a Maple function: "BesselI(0,x)". Evaluate f and "BesselI(0,...)" at several points to convince yourself that they are the same.

**(b) ** Find the Taylor series of f(x) at x=0
by starting with
the Taylor series of exp.
Hint:

2 Pi / | n | cos (t) dt = 0 if n is odd | / 0 (1 - n) 2 pi n! = -------------- if n is even 2 ((n/2)!)

**(d) ** Study the convergence of the Taylor series
to BesselI(0,x), using the methods we used for exp in
Lesson 23. Are the results similar?

**(e) ** The function f(x) satisfies the differential equation
x f''(x) + f'(x) - x f(x) = 0.
What is x S[10]''(x) + S[10]'(x) - x S[10](x),
where S[10] is the Taylor polynomial of degree 10?

**4.(a) ** Find the Fourier series for the function f(x)
defined on [-Pi, Pi] by f(x) = 1 for x > 0 and -1 for
x < 0.

**(b) ** Produce an animation of the partial sums, so that the
Gibbs phenomenon can be seen. Near what points does this occur?

**(c) ** Show that the ratio of the size of the "overshoot" to
the size of the jump is the same as it was for the example we
studied in Lesson 26.