Math 210 Sec. 201 Assignment 4 - due March 27, 1996

1.(a) Consider the integral ∫_{0}^{1} \sqrt{x} dx, with ME(n), T E(n) and SE(n) the errors for Mid-

point, Trapezoid and Simpson’s Rules. Plot n^{3/2}ME(n), n^{3/2}T E(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 ∫_{0}^{∞} \frac{dx}{x + \sqrt{x^{3} + 1}} into 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 C = ∑_{i=0}^{∞} \frac{(−1)^{i}}{(2 ∗ i + 1)^{2}} . By 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: C = ∑ j = 0^{∞}(a_{2j} + a_{2j+1}).

(b) Euler’s constant γ is defined by γ = lim_{n→∞} (∑_{i=1}^{n} 1/i − ln(n)). By finding bounds on

the tail of the convergent series ∑_{i=1}^{∞}(1/i + ln(i) − ln(i + 1)), determine the value of γ

with an error less than 10^{−8}. Compare to Maple’s “gamma”.

3.(a) Let f(x) = \frac{1}{2π} ∫_{0}^{2π} exp(x cos(t)) dt. Maple 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:

∫_{0}^{2π} cos^{n}(t) dt = 0 if n is odd and 2^{1−n}πn!/((n/2)!)^{2} if n is even.

(c) Check your answer for (b) by using “taylor” on f(x) and BesselI(0,x).

(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 xf′′(x)+f′(x)−xf(x) = 0. What is xS′′_{10}(x) + S′_{10}(x) − xS_{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 [−P i, P i] 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.