# ASSIGNMENT 2.3

```ASSIGNMENT 2.3
There are two parts to this assignment. Part A is to be completed online before noon on Monday, January
31. Part B, which should be typeset (preferably with LATEX), is to be submitted to the assignment box by the
same time.
Include your full name and student ID, along with the subject and assignment number, at the top of the front
page.
Part A [15 marks]
This part of the assignment can be found online, labelled Assignment 2.3, at www.mathxl.com. Questions
about the MathXL program itself should be sent to Eric; in fact, questions sent through the site will be directed
to him. To ask me questions, email me directly or come to my office.
Part B [15 marks]
1. (a) Evaluate
X (2 − 3x)n
at the right endpoint of its interval of convergence.
n
n≥1
(b) Evaluate
1 2
3
4
+ +
+
+ &middot;&middot;&middot;.
3 9 27 81
1 1 1
2. Prove that π = 4 1 − + − + &middot; &middot; &middot; . (Hint: find a power series representation, along with its interval
3 5 7
of convergence, for arctan x.)
3. Recall that in the first term we developed methods for solving differential equations of the form
ay 00 + by 0 + c = 0,
where a, b, and c are constant coefficients. However, most differential equations do not have constant
coefficients. One example is the Bessel equation
x2 y 00 + xy 0 + (x2 − 1)y = 0,
(1)
which arises in problems of wave propagation. Here, new methods are required. Find one nontrivial
solution to (1) of the form
X
y=
an x&micro;+n .
n≥0
```