Math 1B
Project # 3
20 Points
Due Date: Monday 9 May
NAME
Series (11.2-11.3)
This project may be done in groups of 2 – 4 students. Each group will submit only one set of
solutions but each group member is expected to contribute. Please show all work on separate
sheets of paper and staple this page to the front.
1. A right triangle ABC is given with ∠A = θ and AC = b .
CD is drawn perpendicular to AB, DE is drawn perpendicular
to BC, EF ⊥ AB, and this process is continued indefinitely, as
shown in the figure. Find€the total length of all the
€
perpendiculars
€ CD + DE + EF + FG + ...
in terms of b and θ .
€
2. Find the€value of c such that
∞
∑e
nc
= 10 .
n=0
3. The Riemann zeta-function ζ is defined by
∞
1
x
n=1 n
€
ζ ( x) = ∑
€
and is used in number theory to study the distribution of prime numbers. What is the domain
of ζ ? (For series, the domain is the set of real numbers x such that the series is convergent.)
4. Find all values of c for which the following series converges.
€
∞
⎛c
1 ⎞
∑ ⎜⎝ n − n + 1 ⎟⎠
n =1
∞
5. Consider the series
n
∑ (n +1)!
n=1
a) Find the partial sums s1, s2, s3, and s4. Do you recognize the denominators? Use the
pattern to guess a formula for sn.
b) Use mathematical induction to prove your guess.
c) Show that the given infinite series is convergent and find its sum.