PHZ 3113 Fall 2009
Homework #2, Due Friday, September 10
1. Write the first five terms of the series for sin[(1 + x)1/3 − 1]. Use logical reasoning or
common sense to decide what is the radius of convergence of this series expansion.
2. A function p(x) is defined as
3 sin x − x(3 + x2 ) cos x
.
x4
What is the behavior of p(x) as x → 0? As x → ∞? Sketch the function p(x).
p(x) =
3. A function f (x) has the series expansion
f (x) =
∞
X
n xn
.
(n + 1)
n=1
Write f (x) in closed form (sum the series).
4. The functions sinh x and cosh x are defined as
1
1
sinh x = (ex − e−x ),
cosh x = (ex + e−x ).
2
2
Write the first four nonvanishing terms of the series expansions for sinh x and cosh x. Write
the series for 1/ cosh x. Use this obtain the first four terms of the series for tanh x.
5. A rope with length one foot longer than the circumference of the earth is looped around
the equator and stretched tight from one point. What is the height of this point above the
earth’s surface? Take R⊕ = 4000 mi, take 1 mi = 5000 ft.