# 3.8Assignment 4 - due March 27, 1996

1.(a) Consider the integral
```                 1
/
|   1/2
|  x    dx
|
/
0
```
with 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)
0
```
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

```             infinity
-----          i
\         (-1)
C =    )     ----------
/               2
-----   (2 i + 1)
i = 0
```
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:
```       infinity
-----
\
C =    )     (a   + a    )
/        2j    2j+1
-----
j = 0
```
(b) Euler's constant gamma is defined by
```                             /  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 = 1
```
determine 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    |
/
0
```
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:

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